To begin with, observe that the notion of a rectifiable curve makes sense in, say, a smooth or a PL-manifold but not in merely a topological manifold. Indeed if $f:[0,1]\rightarrow U\subset{\Bbb R}^n$, $U$ open, represents a rectifiable curve in $U$, and $g:U\rightarrow V$ a mere homeomorphism, we would generally have no reason to expect a rectifiable curve from the composition $g\circ f$. 

Certain homomorphisms (e.g. smooth, PL) do preserve rectifiable curves; the totality of these forms a pseudogroup and we may use this pseudogroup to equip manifolds with just enough geometric structure to distinguish a class of curves as rectifiable.

Do manifolds with just this much structure (I'll call it a rectifiability structure till I know better) occur in the literature? And the usual questions: do all manifolds support a rectifiability structure? if one exists, is it unique up to homeomorphism?

I'd be interested in similar questions relative to higher dimensional objects - manifolds where embedded $n$-disk "have" an $n$-volume ("have" in scare quotes because I only mean relative to each patch of any atlas).

Just a final remark: biLipschitz homeomorphisms preserve rectifiability (right?), but not all rectifiability preserving homeomorphisms are biLipschitz (e.g. $x \mapsto x^{1/3}$), so if I've reinvented the wheel, I haven't reinvented *that* wheel.