Comment by jepler
You're right that the hilbert curve only visits certain points in the unit square, and never a (non-rational,non-rational) point. While the Wikipedia article doesn't seem to mention it, other sources like [1] mention that the definition of a space-filling curve is one that comes arbitrarily close to any point within its space. I think you would be able to see that the iteration of the hilbert curve does get arbitrarily close to (say) the point (sqrt(2)/2, sqrt(2)/2).
The Hilbert curve does contain every point in the unit square. It is a limit of curves, and so can contain points even not in the intermediate constructions. This is similar to how the limit of 1/x as x -> infinity can be 0, even though 1/x never equals 0.