This is the answer to the original question (Not the one which is posted now).
Look in two papers, mine and the paper of Enrico Le Donne.
You are looking for spaces which admit length-preserving embedding into Hilbert space.
In my paper I prove that a compact length spaces which (roughly) admit a length-preserving map into Euclidean $m$-space has to be inverse limits of $m$-dimensional polyhedral spaces.
The infinite dimensional case is easier; it can be done along the same lines; in this case the dimension of polyhedral spaces will go to infinity. It seems that if a compact space admits a length-preserving map into infinite dimensional Hilbert space then it can be perturbed into length-preserving embedding. Enrico considers length-preserving embedding in finite dimensional case (which is much harder).
About the last question. Nash's theorem can be indeed simplified considerably if the target space is infinite dimensional. You can apply the same procedure as in the proof of Nash--Kuiper theorem, but since the dimension is infinite, you can use a parallel frame each time. This way you get a sequence of maps $f_1,f_2,\dots$ which converges to isometric embedding and such that the $k$-th coordinate of $f_n$ is the same for all large $n$. This way you can get better regularity of the limit --- instead of $C^1$, you should get $C^\infty$ (am I right?).