Javascript, 34 bytes

for(r=[b=[i=10]];i--;r[i]=b)b[i]=1

Since it's apparently OK to make the rows equal by reference, I guess it's apparently OK to rely on that fact. This helps us shave off one for-loop by building the table at the same time as its rows. So, here's my new contestant.

Since r[0] and b[0] are overwritten during the loop, they can contain garbage. This gives us another free execution slot to shave off some commas. Sadly, r=b=[] won't do, since then they are equal by-ref.

Also, it scales well (99x99 is still 34 bytes), doesn't require ES5 (the new functions have terribly long names, anyways), and as a bonus, it's unreadable :-)

ECMAScript 6 - 33 Characters

x=(y=[..."1111111111"]).map(x=>y)

Outputs a 10x10 array of "1"s.

This abuses the fact that the string "1111111111" has all the requisite properties to be treated as if it is an array so you can use the spread operator ... to transform it into an array of characters and then map it to a copy of the array with each element referencing the "original".

Or with only one variable name (for 35 characters):

x=(x=x=>[..."1111111111"])().map(x)

Or for extra confusion (but at 45 characters)

x=[];x[9]=x;x=[...x].map(y=>[...x].map(x=>1))

or (43 characters)

y=y=>[...x].map(x=>y);x=[];x[9]=x;x=y(y(1))

44 bytes

for(a=[i=10];i;)a[--i]=[1,1,1,1,1,1,1,1,1,1]

Previous version:

for(a=i=[];i^10;)a[i++]=[1,1,1,1,1,1,1,1,1,1]

45 characters

x=[a=[1,1,1,1,1,1,1,1,1,1],a,a,a,a,a,a,a,a,a]

Each element points to the same array, but the spec doesn't mention anything against that!

Javascript, 51

a=[];for(i=10;i;i--,a.push([1,1,1,1,1,1,1,1,1,1]));

Or if all indices are allowed to point to the same array:

Javascript, 41

a=[1,1,1,1,1,1,1,1,1,1];a=a.map(a.sort,a)

JavaScript, 45 44 Bytes

Best I have so far. (Still trying).

x=[];while(x.push([1,1,1,1,1,1,1,1,1,1])-10);

Shorter (thanks mellamokb!)

for(x=[];x.push([1,1,1,1,1,1,1,1,1,1])-10;);

Javascript, 47

Since my original solution has been beaten, I will now post it.

for(a=[],i=10;i--;a[i]='1111111111'.split(''));

Unfortunately, 0x3FF.toString(2) isn't quite as efficient as just listing the string out, which isn't quite as efficient as just statically declaring the array.

You can shave off one character this way (46):

for(a=[],i=10;i--;a[i]=[1,1,1,1,1,1,1,1,1,1]);

You can save 2 more characters like so: (44)

for(a=[i=10];i--;)a[i]=[1,1,1,1,1,1,1,1,1,1]

Another 44 byte solution using JS 1.8 function expression closures (FF only atm):

x=(x=[1,1,1,1,1,1,1,1,1,1]).map(function()x)

JavaScript, 57 bytes

r=eval("["+(A=Array(11)).join("["+A.join("1,")+"],")+"]")

Before golfing:

a=Array(11).join("1,");
b=Array(11).join("["+a+"],")
c=eval("["+b+"]")

Note: This needs ES5, so don't expect much from IE.

54

This has already been beaten, but here's my solution:

function x(){return[1,1,1,1,1,1,1,1,1,1]}x=x().map(x)

39 bytes

Using array comprehension in ECMAScript 7:

x=[x for(y in x=[1,1,1,1,1,1,1,1,1,1])]

56 characters

Rather long answer, but will be better for 100x100 etc.

for(a=[[i=k=0]];++i<100;m?0:a[k=i/10]=[1])a[k][m=i%10]=1

Javascript, 36 chars (ECMA 6)

(a=[b=[]]).fill(b.fill(a[9]=b[9]=1))

Tested in my browser (Firefox on Ubuntu) and in the Codecademy Javascript interpreter

ES6, 54

It's not the shortest, but I thought I'd go for a different approach to what's already here.

a=(''+{}).split('').slice(0,10).map(_=>a.map(_=>1))

Cast an object to a string "[object Object]"
Split the string into chars ["[", "o", "b", ... "]" ]
Grab a slice of just 10
Map the result of mapping 1 onto the initial slice

61 for fun

eval("r=["+("["+(Array(11).join("1,"))+"],").repeat(11)+"]")

Javascript - 64 63 59 characters

for(var a=b=[],i=100;i-->0;b[i%10]=1)if(i%10==0){a[i/10]=b;b=[]}

for(var a=b=[i=100];i-->0;b[i%10]=1)if(i%10==0){a[i/10]=b;b=[]}

One char saved by hiding the iterator in the array.

for(a=b=[1],i=100;i--;b[i%10]=1)if(i%10==0){a[i/10]=b;b=[]}

Saved some chars with Jans suggestions.

Readable version:

for(var a = b = [i = 100]; i-- > 0; b[i % 10] = 1) {
  if(i % 10 == 0) {
    a[i / 10] = b;
    b = [];
  }
}

Yes, it's not the shortest solution, but another approach without using readymade [1,1,1,1,1,1,1,1,1,1] arrays.

Javascript 69 characters (for now)

b=[]
a=[]
for(i=100;i-->0;){
a.push(1)
if(i%10===0){b.push(a);a=[]}
}

JavaScript 73

function h(){s="1111111111",r=[10],i=-1;while(++i<10){r[i]=s.split("");}}

Javascript ES6, 53

x="1".repeat(100).match(/.{10}/g).map(s=>s.split(""))

But the ones in the array are strings :)

Javscript ES6, 53

x=eval("["+("["+"1,".repeat(10)+"],").repeat(10)+"]")

JavaScript (ES6), 35 bytes

A=Array(10).fill(Array(10).fill(1))