  The $\mathbf{O}(D,D)$ covariant generalized metric, postulated as a truly
fundamental variable, can describe novel geometries where the notion of
Riemannian metric ceases to exist. Here we quantize a closed string upon such
backgrounds and identify flat, anomaly-free, non-Riemannian string vacua in the
familiar critical dimension, $D{=26}$ (or $D{=10}$). Remarkably, the whole BRST
closed string spectrum is restricted to just one level with no tachyon, and
matches the linearized equations of motion of Double Field Theory. Taken as an
internal space, our non-Riemannian vacua may open up novel avenues alternative
to traditional string compactification.
