I can't comment on Noah's answer, so:

The reason for the C^1 condition is that [Nash's C<sup>1</sup> embedding theorem][1] says that any Riemannian k-manifold with a short embedding into R<sup>n</sup> has an isometric C<sup>1</sup> embedding into R<sup>n</sup> for any n > k. In particular, there is an isometric C<sup>1</sup> embedding of the hyperbolic plane into R<sup>3</sup>. The very beautiful proof proceeds by "pleating" the embedding to be closer and closer to isometric, while keeping it contained in a not-much-larger region.


  [1]: http://en.wikipedia.org/wiki/Nash_embedding_theorem