The kicked rotator on a torus is a system with a bounded phase space in which a chaotic diffusion occurs for a large enough perturbation strength. The quantum version of this model exhibits localization effects which produce deviations from random-matrix-theory predictions. We show that these localization effects display a scaling behavior which is a counterpart of the scaling theory of one-dimensional Anderson localization in finite samples. We suggest that this behavior can be highly relevant to some general problems of quantum chaos.