Motivated by the possibility of a finite theory of gravity provided by superstrings in ten space-time dimensions, we analyze the problem of space compactification in the context of string dynamics. Such an analysis is hampered by conceptual and technical problems, stemming from the existence of the quantum string's own graviton mode on the one hand, and from Witten's observation of anomalies in a not specially chosen curves space-time on the other hand. Still, in the context of a classical local field presentation of string theory à la Nambu and Hosotani, supplemented by gravitational and Kalb-Ramond interactions, we are able to find solutions with space compactification. It is the antisymmetric tensor zero modes that dictate this compactification towards three space-time dimensions for ordinary strings or towards four or five space-time dimensions for superstrings.