We demonstrate that migration away from self-produced chemicals (chemorepulsion) generates a generic route to clustering and pattern formation among self-propelled colloids. The clustering instability can be caused either by anisotropic chemical production, or by a delayed orientational response to changes of the chemical environment. In each case, chemorepulsion creates clusters of a self-limiting area which grows linearly with self-propulsion speed. This agrees with recent observations of dynamic clusters in Janus colloids (albeit not yet known to be chemorepulsive). More generally, our results could inform design principles for the self-assembly of chemorepulsive synthetic swimmers and/or bacteria into nonequilibrium patterns.