You have to be careful when talking about unqualified 'size', though - while the naturals and rationals have the same cardinal number and thus have the same size in that sense, there is also a sense in which one is clearly bigger than the other. The naturals are a strict subset of the rationals, which is a reasonable sense in which they're strictly smaller in size.
Think of it this way:
We have two infinite sets: A and B. Let us imagine a process (isn't that word loaded after reading this thread...) wherein B systematically takes its elements and brags about them to A. Once it uses an element, it will never show it again.
If A, in turn, can counter anything B shows with an element of its own that it has never used before, then:
|A| >= |B|
And, if we reverse this process, with A showing its elements to B, and B counteracting, then we have:
|B| >= |A|
Well, if both of those things are true, the only alternative is:
|A| == |B|
THIS IS NOT A PROOF. This is a way of thinking.
The problem people have is this thought:
"If every element in A is also in B, but B has elements beyond that, then B must be bigger."
This is a symptom of finite sets, not an actual rule. We equate them in our heads because there's a 1-to-1 relationship in the finite space, but they are not the same.