A recent proposal suggests that experimental discrepancies on the lifetime of neutrons can be resolved if neutrons decay to dark matter. At the same time it has been demonstrated that such a decay mode would soften the nuclear equation of the state resulting in neutron stars with a maximum mass much below currently observed ones. In this Letter, we demonstrate that appropriate dark matter-baryon interactions can accommodate neutron stars with mass above two solar masses. We compare this stabilization mechanism to one based on dark matter self-interactions, finding that it is less sensitive to the details of the nuclear equation of state. We present a simple microscopic model realization of this mechanism.