There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from.
From my understanding, a Hilbert curve can completely fill a plane, meaning it runs through every single point on that plane. This seems like to me like a very closely related concept to finding a way to list all of the real numbers, as both creating a list of the reals and completely filling a space with a curve seem impossible for the similar reasons. Yet one turned out to be possible, so in my head at least that would imply that the other should be as well.
Again, my understanding of Hilbert curves is too superficial to construct a very specific question, so the best I can hope to do is outline a general description of the feeling I'm trying to get at, and hope that more educated users are able to pick up on what I'm thinking and explain what I'm failing to consider.
Edit: I think what the real question I'm trying to get at is, why are these two concepts not both impossible for the same reason?