The Black Death - bubonic plague - has a mortality rate of 50-70%, so even with no other measures there's a pretty good chance of any two people surviving at random (about 25% if you take the lower end).

However, population-wide statistics are misleading. The mortality rate will be very much lower for fit, healthy people who are well-fed and have no underlying health conditions. So the survival rate for your two could be very much higher.

Also, genetics come into it: if your family has higher natural resistance than both, then both people will benefit.

Finally, let's cheat. Their father is a gravedigger, so they have received inoculating doses of a wide variety of conditions which have a synergistic effect of boosting their immune systems against this particular pathogen, so they suffer no more than a minor sniffle and are good for the next generation.

Black plague (and hopefully any disease which closely copies it) is carried by fleas. Fleas hate citrus fruit, especially lemons. The woman and her husband own the only citrus grove near the town and normally make their living selling lemons.
So their home and grove would be a natural safe zone from black plague. If only her husband hadn't joined the death teams, he would have survived too.

Bonus Points : In addition to being copper-rich, Altheia is also the empire's primary source of citrus fruits. The main character's distant ancestors were from Altheia which is how she ended up with the family citrus grove in the first place.

If you don't like citrus, there are a number of other natural flea control methods which would serve. Here is an article on the subject.

They live in the trees.

She knows that the Death can jump from person to person. And she knows that even without the Death, people have become dangerous. Even among the uninfected, people are desperate and some are insane. Before he went away her husband told her some of the things he saw in the houses of the dead, and would not tell her about some others.

She goes to the forest, and she goes up. When she was a child her mother showed her secret places high in the trees, and now she brings her own children to the refuges she once knew.

The three of them live quietly, high above the ground. Beech and oak are generous with their gifts, and as a child the woman learned other things that the forest offers to eat. People do not come there. Nor do rats, or fleas. And from their vantagepoint the family is witness to mysterious, terrible and beautiful things in the forest around them and on the ground below.

One solution is to have them come very, very close to death, but then just barely pull through. It's easier to believe "these two people almost died of the plague" than "these two people were immune to the plague" or "these two people never got the plague". Sometimes people do simply recover from even the worst conditions.

The mother is a cat hoarder. She adores cats, and cannot have too many of them. Any rat looking for living quarters takes one sniff near her house and decides it would not make a good home.

The cats are kept mainly indoors, well fed, and groomed regularly. She does not like them getting fleas. If she sees fleas on a cat she combs it thoroughly. She insists everyone living in the house must bathe and wash their clothes regularly.

The resulting low exposure to fleas greatly reduces the risk of the children being bitten by infected fleas.

Cosplay the doctors

As the image shows, while some of their measures and habits were simply useless, some of their decisions in what to wear and what to do were correct. Their uniforms, while not perfect, did provide them with decent protection both against the sick patients' bodily fluids and against the fleas which transmitted the disease. To maximize their chances, steal this look (minus the mask and hat, those are important in identifying the real doctors, especially the hats, and might cause you trouble to wear), making sure to cover as much of their bodies as possible with protective clothing, and cover everything with some leather cloak or other garment that hides the doctor outfit as to not raise suspicion. It won't 100% guarantee that they'll reach the destination, but it sure will give them an edge when it comes to avoid contamination by fleas and fluids alike.

Now that is just precautions and safety measures to increase their chances,but it won't be enough. The kids must watch out not to raise suspicion and keep away from urban centers, in order to avoid both the agglomeration with potential infected and the unsanitary environments which cities were. They also must watch out what they eat and drink, avoiding as much as possible dubious sources along their path.

So summing up:

• watch out for what you eat and drink
• copy the doctors'uniforms, they're doing it right (kinda)
• stay as far away from cities, villages and people as a whole as possible.
• stop as little as possible on your journey, so long as you don't violate the previous recommendations. The sooner you arrive, the better, but don't go skipping too many nights, as lack of sleep will affect your immunity.
• pray that these measures are enough and may the Lord guide you through your journey.

Obviously if your character has no idea where to go, the only answer she'll be able to manage will be "not here" and "away from people". As it happens, historically, quarantining in the countryside is exactly what the wealthy did and it sometimes worked. You can check out footnote-heavy copies of Boccaccio's Decameron for details.

Really, though, if they weren't wealthy enough to hole up at a country estate and send servants to pick up supplies, they'd be moving from place to place and trying to pick up info on rumors. There would doubtless be many floating around, some nonsense, some genuine information, some traps laid by bandits and slavers.

Some states would be largely immune through having quickly quarantined upon trusted reports arriving of the mess elsewhere. Any place successful, though, would have policies in place precisely to keep refugees like your characters from being able to come in.

Assuming that your disease uses the same mechanism as the Black Death, fleas can't physically handle sulfur. If they were in a highly sulfur-rich environment like a mine or volcano they'd be fine aside from the molten rock. Sulfur was also used since antiquity by some farmers as an insecticide and by some textile workers in dyeing and finishing processes. Any wouldbe alchemists or gunners would also hoard the stuff. Your characters could end up in one of those trades, with some of those workers, or (safest and most sensible) hiding in the hideously stinky supply rooms and warehouses that store the stuff. They could go there just to avoid other people in the most natural way possible (find somewhere nonharmful that is still so awful no one else would come near) and end up the better for it.

You can always change the mechanism for your disease, though. Person-to-person transmission makes quarantine/hiding the obvious solution. You could come up with specific virus qualities that they wouldn't understand but that would match wherever you need them to be as part of your story, though. Maybe high altitude or high humidity or complete aridity or high salt concentrations could neutralize whatever the illness happens to be. Etc.

The disease has an age-specific mortality rate which increase kids' survival chances

If you haven't already written a lot about your infectious disease already, you might consider to make it more dangerous and lethal for the adult people, especially the age cohort which already has teenage children.

Many viral infections are much more mild for the kids, the best-known example being chickenpox, but also COVID-19 now, and although real-life plague didn't (and doesn't) have such a property, you can make up a fictional disease which is lethal for a small fraction of preadolescents (a fraction which is negligible in the context of medieval children mortalities), a larger fraction of teenagers, and then its danger increases with age. In such a case the death of the father and survival of the children will be perfectly natural.

You will have to design symptoms different from the plague for your disease though! Another possible disadvantage is that I won't expect a viral disease like that to have a CFR as high as the plague (~30—50%) but rather more like Spanish flu (~10%).

Your characters live in a sparsely populated area.

• The land is sparsely populated, most live on subsistence farming, there is less need for widespread trade
• Settlements are far apart, thus any infected traveller is likely to become sick during travel, hence they die on their way, or shelter in monasteries until they are fit again.

There's a question over a history.se, which asks the question why Poland was pretty much spared

I liked the citrus farming answer, but perhaps extend it a bit. Make citrus juice in large quantities by squishing it with their feet, thereby getting it all over themselves on a regular basis. If people think it is a cure to the plague, there might be high demand for it, therefore they would be making it regularly. Use citrus candles regularly because they like the scent, thereby helping keep the flees out of the home.

Some feral cats on their farm could hunt the rats down. Of course, cats carry flees too, but there would be fewer, and it would be easier to keep them out of the house.

Copper is anti-bacterial. Perhaps they can have a side business of copper or bronze smithing? I read an article recently that coppersmiths died in far fewer numbers during the plague than the rest of the population. Sorry, I could't find the article in my 5 minutes of searching. I think it had to do with them breathing in copper dust on a regular basis.

The farm is naturally "socially distant". The family doesn't necessarily need to interact with many people, other than to get food and supplies. They can use a middle-man who sells their products for them, and perhaps brings them their supplies, thereby limiting exposure even more. Of course, this person might be a carrier. People understood that being close to sick people made it more likely for them to get sick, so the seller might wear a mask (they understood this principle as well) and possibly pickup/dropoff at a distance.

Also, the death rate of the plague was much higher in Southern Europe, which are the numbers you usually see (2/3rds of the population). In Northern Europe, the percentages were much lower. I don't recall if this was due to colder climate or population density. If colder climate, I don't recall if it was because the plague didn't transmit as easily, or if people just naturally don't get as close to one another and wear more clothes (it's been well over 20 years since my Western Civilization courses in college). Of course, colder temperature would probably preclude a citrus farm.

The Plague was spread by fleas. So make your people very clean and antisocial.

This actually happened in history, there was a religious group whose customs involved washing a lot, and they survived better.

Of course the neighbours got suspicious and behaved like ignorant people, but you can find a way to handle that...

Start with more

People didn't have two kids and raise them to adulthood in that period, they often had as many as they could in the hope that some survived to adulthood. Surviving childhood was a major life achievement.

If you want two characters to survive the plague, start with a whole family, six to ten people, parents and children, kill all but the two you want.

Perhaps that's not the story you want to write, but it'll bring home how deadly your plague is. You've already killed the father, and presumably the mother will go soon, add a couple more kids and you've hit average. Only one or two surviving from a family is reasonable and nobody will question it.

Do you need these two specific people to survive, or do you just need two people who did survive?

What I mean is, out of a world with many families, by pure chance there will be some families where you have two survivors that match your criteria. You can just tell the tale from their perspective.

Make It Interesting

How do they survive? Well, as many of the people above have said, they could just randomly survive or come really close. to death, but since we're discussing Worldbuilding, for some kind of media, it would make sense to make it interesting.

I'm not necessarily saying that the others' answers are wrong, but the audience doesn't want to hear how they randomly survived. No, they want to hear something thrilling, something interesting. So you can go about this two ways:

1. Quarantine In The Wild

You can make the main characters camp out in the wilderness, away from other people, struggling to survive. Narrate their struggles, then after a few weeks, have some stranger affected with the Black Death approach them, of course, they will remain up in their base while he mumbles about some place called Altheia, he will tell them its location and ask them to help him. The brother will hesitate, but the sister will tell him to run, and when his back is turned, she will shoot him with her bow. Then, then the two will journey to Altheia and will settle there.

2. Wandering

Have the siblings wander to the north, based on rumors of a safe haven that they heard about around town, finding shelters and food wherever they can find it, barely surviving on what they find. Eventually, the two are about to starve when they reach an inn, they don't want to enter, but they have to. There, they find that they are in the outskirts of Orotheia (change the name to whatever you wish), about 12 miles from Altheia. The siblings join a caravan of Altheian traders heading back to Altheia. However, their navigator gets killed by a bear, so they accidentally wander into the part of Altheia where the infected. are held. They come under attack. The rest of the caravan is infected and killed, but the siblings escaped just before the caravan came under attack. They silently move through the forest, eventually, they manage to reach Amaltheia and live to tell the tale.

This theory is similar to everything else and has a similar result, but it's likely more appealing and interesting to read/watch/view, which makes the entire thing all. the more interesting and enjoyable.

Hope this helps!

The mother gives the children in adoption to a noble person who lost their own children.

The noble would take extra care of the children by nourrishing them, clothing them and attending to their hygiene and even without the exact knowledge of preventive measures incidentally avoiding the tragedy repetition.

To intensife the chances you might want to use the fact that according to a recent study the primer strain of the plague was likely transmitted by food and not fleas during a time similar to that in your story (so clean and fresh food would be enough to keep the children safe).

Put all the chances on their side

Many answers are good here, so you could combine them for your characters.

For instance:

• From david-hambling answer: Their family has a good genome for resisting the plague, and the father was a gravedigger that helped boost their immune system.
• From Henry Taylor answer: They enjoy citrus.
• From Brian answer: the cats
• Inspired by RedSonja answer: Their religion has a strict standard about washing hand and mouth before eating. It is possible that is one of the cause, among other factors, that jews were a bit spared during the black plague.
• From aim92 answer: They're young and healthy in the first place.
• Also a recurrent topic among answers is social distancing.

Putting it all together

So build characters that group some of these characteristics. For instance:

Young person, working in a small group as a sentinel at an outpost for extensive periods of time. To get to the outpost, there is a longer than incubation time travel from the dispatch center. There are cats near the outpost.

The outpost tower is located in an area with lots of citrus. When the characters travels, he/she always take some citrus from the outpost in its bag. His/her rank in the army is provided with individual room and a (big) horse, so that he/she is sitting well above the crowd in denser areas.

Father was a gravedigger for the army, which kept them well fed. Due to its job making his hands particularly dirty, the father installed a small basin outside the family house to be able to wash his hands and face when coming back from work just in time to eat. So the kids naturally imitated it and kept washing hands and face before eating in their adult life.

The character has a birth mark on the face, lots of people will keep distances by reflex. Sometimes, to hide it, the character wears a cloth mask.

But

Do not explain that any of these saved the life of your characters. A careful reader will understand that the characters did survive because of a combination of factors. On the other hand, from the characters point of view, it will not be possible to understand why they were spared. It should be self-explanatory for the reader, not for the characters.