Well, the first thing you need to worry about is sustaining life, that means farms, agriculture, and given trade will be impossible(cannot leave island), you need to make enough to have a buffer for bad times.

So, what was the basic rule? An acre of farmland per person. And that's just survival. Given trade is impossible, you may want to invest more in backups--so I'll go with an acre and a half per person.

So, now we have the basic calculation of how much per person. How many people are needed to ensure inbreeding doesn't become a problem? That's a bit more problematic, given it's more about monitoring who makes babies with who. But as a rule of thumb, let's go with an upward of 3,000 people (check around on this site, I'm possitive I remember a question that deals with that, something about a generational mothership meant to find a new habitable planet)

So, 1.5 acres per person. 3,000 people. 4,500 acres just to sustain the people. Now, you need resources beyond survival. You need building materials, you need metal ore, you need industrial terrain to process this, and you need space to train people to do so. You also need government to guide and govern the people, and you need space to do research. (yeah, this isn't quite as simple as your originally worded question makes it)

So, given resources are limited, you cannot just throw anything away. Once someone finds a mining hotspot, and you start mining for your metals, you're going to have a LOT of useless dirt and rock and slag and other junk you have no use for. Find a use, or your people will suffer in the long run. Use it to build houses, for all I care. But given you need a metaphorical global economy on an island? Waste is your enemy.

And that includes the waste of the people! Both sewage and garbage, you'd better have something to combat that, or your people will be facing their own Black Death. Avoid that at all costs. Doubling the fact that these people will be dependant on fisheries, unless there are enough farm animals for meat and the like, you'll need them to be super careful with overfishing (given the limited range of where they can fish). That means they cannot pollute the water, and they cannot chase away the fish they need to come back. This might sound like a joke, but check Moana for island-locked life and what it takes to ruin that forever. Nothing can or should go to waste. Nothing.

So, with this additional information, we need a basic idea of how much space for the extra needs. Let's round it off at 3 acres per person, for all the entertainment venues, government, research facilities, and the like. Keep in mind that once technology starts becoming more predominant they can start building up. Or just start building smarter! Harappan civilization might help you there. They were excellent builders, and city planning seems to be at the forefront of their planning.

Combine all this, and you'll have a civilization that might go the distance. But you're not likely to get any further than European middle ages, simply for their lack of trade (personal opinion, and treat that as such).

Smallest real-life examples

This is a list of mostly isolated 'islands' and how far along they got technologically. It is relevant that we don't ultimately know how far these islands would have progressed if those silly Europeans hadn't shown up with their guns, germs, and steel and made everyone wear pants. But this post isn't about speculation, its about history.

Sustained hunter-gatherers:

• Chatham Island (966 km$^2$) (smallest that I could find, there may be smaller)
• The tragic story of the Moriori people, residents of Chatham island, is that they emigrated there from New Zealand in ~1500 AD. Since their crops wouldn't grow in the sandy soil they reverted to hunter-gatherers and lived in peace for the most part. 300 years later, a group of their Maori cousins from New Zealand decided to conquer the island. They kidnapped the first mate of a British brig, and made the ship take 900 of them to Chatham Island where they proceeded to massacre and enslave the ~2000 Moriori. In any case, this is an example of an island being small enough to enforce a hard limit on advancement. Easter Island is a similar case.

Sustained pre-metalworking chiefdoms:

• Polynesia (~30,000 km$^2$)
• I'm not counting New Zealand, since the chiefdoms had already developed on Tonga and the Societies before New Zealand was settled around 1280.

Sustained bronze-age culture:

• The South America (17,840,000 km$^2$)
• Of course, South America is connected to North America. But it is interesting to note how much North America brought to the table: maize. That's it. Pretty much all the rest of the developments in agriculture, husbandry, and metalworking came from South America, along with the most advanced construction and government systems. North America (the Maya) did make the first/only writing system, though there is some debate as to whether quipu is a writing system or not. In conclusion, South America probably would have gotten as advanced as it did by itself.

Sustained Age of Discovery culture:

• Eurasia (54,759,000 km$^2$)
• Again, while Africa is connected, and North Africa and Egypt were an integral part of technological development, I think it is a fair assumption to say that most agriculture and technology would have advanced at about the same rate if Africa wasn't around. It is also reasonable to assume that Eurasia could have made it to modern technology without the rest of the world, as well, though perhaps not as quickly.

The most important factor for technological advancement is the understanding of underlying principles. You might have some advanced technology, but if you don't understand how to replicate, change and improve it, it's not much more than a gimmick.
The Romans had 'steam engines', but they had no useful understanding of air so it wasn't until we understood thermodynamics before we could design steam engines that were actually useful.

For understanding you need a lot of people that have nothing better to do than trying to figure out how stuff works. (If you already have a blue-print, depending on how well it teaches your the understanding and how well you can read it, that's a lot easier.) The speed of your technological advancement scales with the number of people 'doing science'. Which, if that's all you want to do, is the biggest number of people the society can afford to not produce food.

If you advance your technology, it'll require more education to reach the current level of understanding, so overhead (schools etc.) will become a bigger factor. On the flip side, many of the new technologies improve the productivity of the food producing people so you'll need fewer of those.

An island large enough to sustain technological advancement: one large enough to sustain a population. Bigger just means faster.

As per tech level: assuming you've figured out agriculture, until well into the iron age I don't see why an island of a few km in diameter won't suffice, given some luck in available iron deposits near the surface. Then with just iron and trace amounts of other minerals (copper, tin, etc.) you can develop well into the renaissance. Charcoal from wood can replace coal. Again, more area means more trees means faster development.

For more modern technologies you'll also need other more rare elements. Story-wise this is easy to fix, just have the island be a (former) volcanic island with plenty of metal and rare earth element veins. I am no geologist, but I would deem it somewhat plausible that an island the size of, say, Ireland could have enough minerals for more modern technologies.

First of all, You should be branching out onto the waters making more space for your island anyways.
As you said, resource availability is the problem. For technological advance, you're going to need all sorts of resources; An island on its own cannot sustain itself.
If you are able to get the resources from an alternate place, then disregard the above. I think you'll need to define your technological advance for me; Are you building an underground bunker with a new type of nuclear bomb that takes up three entire city blocks? Are you advancing your tech towards some sort of reactor that only takes up a handfull of space?
A small island could be viable depending on the size or type of technology you're building; Do you want specific area's of the island? 300*300[]'d? It's really not easy to say.
Is your technology small or large? What kind of technology are you looking to create?
etc etc Military tech:
I can't imagine you would need that much space for building a new type of tank. If you have a place that you can use to create things, then it should be easy to build new tanks, new firearms, planes, etcetera.
A small island should be sufficient. The island itself could be any size, but the thing you need to worry about is resources. You can build off of the sides of the island no matter what size the island itself is and make more room and space for building.