I can't comment on Noah's answer, so:
The reason for the C^1 condition is that Nash's C1 embedding theorem says that any Riemannian k-manifold with a short embedding into Rn has an isometric C1 embedding into Rn for any n > k. In particular, there is an isometric C1 embedding of the hyperbolic plane into R3. 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.