Accurate characterization of the astrometric errors in the forthcoming Gaia catalogue is essential for making optimal use of the data. Using small-scale numerical simulations of the astrometric solution, we investigate the expected spatial correlation between the astrometric errors of stars as function of their angular separation. Extrapolating to the full-scale solution for the final Gaia catalogue, we find that the expected correlations are generally very small, but could reach some fraction of a percent for angular separations smaller than about one degree. The spatial correlation length is related to the size of the field of view of Gaia, while the maximum correlation coefficient is related to the mean number of stars present in the field at any time. Our scalable simulation tool (AGISLab) makes it possible to characterize the astrometric errors and correlations, e.g., as functions of position and magnitude.