Admitting non-Riemannian geometries, double field theory extends the notion of spacetime beyond the Riemannian paradigm. We identify a class of singular spacetimes known in general relativity with regular non-Riemannian geometries. The former divergences merely correspond to coordinate singularities of the generalized metric for the latter. Computed in the string frame, they feature an impenetrable non-Riemannian sphere outside of which geodesics are complete with no singular deviation. Approaching the non-Riemannian points, particles freeze and strings become (anti)chiral.