Quantities like arc-length/area/volume are invariant under isometries (rigid transformations). In Euclidean space we can define these quantities using integration with respect to Lebesgue measure, because the Lebesgue measure is invariant under isometries.
I think the Lebesgue measure is the only such measure on $\mathbb{R}^n$ with this property -- is this true?
On a more general non-Euclidean space, might there be a measure that's invariant under isometries? And if not, can we still define arclength/area/volume using integrals?
There always exists a measure which is invariant under isometries*, but you'll need to do a bit more if you want to single out the Lebesgue measure as the "nicest" one. For instance, the zero measure and the counting measure are both invariant under isometries on any space. On spaces with fewer isometries, you will get even more options; many Riemannian manifolds have no nontrivial isometries, and so every measure is isometry-invariant.
*On Riemannian manifolds, at least. There there is even a direct generalization of the Lebesgue measure, often called the Riemann measure, which is automatically isometry-invariant.