Engineered dissipation can be employed to prepare interesting quantum many-body states in a nonequilibrium fashion. The basic idea is to obtain the state of interest as the unique steady state of a quantum master equation, irrespective of the initial state. Due to a fundamental competition of topology and locality, the dissipative preparation of gapped topological phases with a nonvanishing Chern number has so far remained elusive. Here, we study the open quantum system dynamics of fermions on a two-dimensional lattice in the framework of a Lindblad master equation. In particular, we discover a mechanism to dissipatively prepare a topological steady state with nonzero Chern number by means of short-range system bath interaction. Quite remarkably, this gives rise to a stable topological phase in a nonequilibrium phase diagram. We demonstrate how our theoretical construction can be implemented in a microscopic model that is experimentally feasible with cold atoms in optical lattices.