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

The reason for the $C^1$ condition is that [Nash's $C^1$ embedding theorem][1] says that any Riemannian $k$-manifold with a short embedding into $\mathbb{R}^n$ has an isometric $C^1$ embedding into $\mathbb{R}^n$ for any $n > k$. In particular, there is an isometric $C^1$ embedding of the hyperbolic plane into $\mathbb{R}^3$. 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]: https://en.wikipedia.org/wiki/Nash_embedding_theorem