It has a boundary, we know (roughly) how big it is, and that it's expanding.
i think it's a decent logical assumption to assume that it has no size (infinite), but i don't really think anyone can prove that. and you certainly can't see or measure all of it. the amount of it which we can see has been measured. but that's not the size of the whole universe, that's just the size we can see, which is known as the 'observable' universe. right now pretty much no one can see further, but that's not a 'space' boundary, it's actually more of a time boundary. looking that far a way looks back in time, and for some reason there is actually a problem with looking back more than 13.8 bil years.... something gets weird and there's like a desktop background of some sort