EDIT

Although I've already accepted an answer, I'll just add some extra context.

My reason for questioning this relates to the Hilbert space-filling curve. I find this interesting because of applications to multi-dimensional indexing data structures in software. However, I found Hilberts claim that the Hilbert curve literally filled a multi-dimensional space hard to accept.

As mentioned in a comment below, a one meter line segment and a two meter line segment can both be seen as sets of points and, but (by the logic in answers below), those two sets are both the same size (cardinality). Yet we would not claim the two line segments are both the same size. The lengths are finite and different. Going beyond this, we most certainly wouldn't claim that the size of any finite straight line segment is equal to the size of a one-meter-by-one-meter square.

The Hilbert curve reasoning makes sense now - the set of points in the curve is equal to the set of points in the space it fills. Previously, I was thinking too much about basic geometry, and couldn't accept the size of a curve as being equal to the size of a space. However, this isn't based on a fallacious counting-to-infinity argument - it's a necessary consequence of an alternative line of reasoning. The two constructs are equal because they both represent the same set of points. The area/volume/etc of the curve follows from that.