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. 



Manifolds with rectifiable curves
This is just Lipschitz structure: only locally Lipschitz maps preserve rectifiability of all curves.


What is wrong with the $x\mapsto x^{1/3}$ map is explained in Tapio Rajala's answer. (A more explicit example is the path $t\mapsto t^2\sin(1/t)$, $t\in[0,1]$, and you can make a non self-overlapping example if you go to dimension 2). Here are the missing details of Tapio's answer in the general case.


If a homeomorphism $f:U\to U$ is not Lipschitz on some compact set, then there exist sequences $x\_n$ and $y\_n$ converging to some $p\in U$ rapidly (e.g. such that $|x\_n-p|<2^{-n}$ and $|y\_n-p|<2^{-n}$) and such that $|f(x\_n)-f(y\_n)|>n^2|x\_n-y\_n|$. Consider a curve that oscillates between $x\_n$ and $y\_n$ approximately $(n^2|x\_n-y\_n|)^{-1}$ times, then goes to $x\_{n+1}$ and oscillates between $x\_{n+1}$ and $a\_{n+1}$, and so on. It is rectifiable (its length is bounded above by something like $2\sum\_n (n^{-2}+2^{-n})$) but its $f$-image is not: each oscillating part adds at least 1 to the length.


