What do we mean by the size of the universe? Isn't it true that the concept of length exists only within the universe? Are we simply measuring the observable universe, that is to say, the spatial limits of our observation? 
 A: It is difficult to say. When you say "The size of the Universe" some people think about the observable Universe, while others, like me, think about the entire Universe, beyond just what we can observe. When it comes to length in the entire Universe, it's useless. the Universe is infinite so no matter what measurement we create, it will be insufficient to measure the entire Universe. In a sense, when people say the size of the Universe, it is easier to assume they are talking about just the parts we can observe.
A: What do we mean by the size of the universe?
Exactly that. We mean what we say.  
Isn't it true that the concept of length exists only within the universe?
Since the universe is everything, yes. But that doesn't stop us talking about the size of the universe. 
Are we simply measuring the observable universe, that is to say, the spatial limits of our observation?
That's what we can measure, but when we're talking about the observable universe we say observable universe. When we're talking about the whole universe, we don't. 
Note that some people say the universe is infinite, see Dragonslayer's answer above, and this NASA article on WMAP. However we have no evidence that it is, and IMHO an infinite universe is at odds with big bang cosmology, which I feel is broadly correct. Search on universe size of a grapefruit and you find articles saying the universe was the size of a grapefruit near the time of the big bang. More recently however, since WMAP found that the universe was "flat", some people have shifted to saying the observable universe was the size of a grapefruit, and that the universe was infinite even when the big bang occurred. I don't buy this myself, because IMHO an infinite universe cannot expand. I'm with Russell McMahon above. What's interesting about the finite flat universe is that it has some kind of edge. But of course, that's one for another day.   
