Look in two papers, mine and the paper of Enrico Le Donne.
Your 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).