Few equilibrium-even less so nonequilibrium-statistical-mechanical models with continuous degrees of freedom can be solved exactly. Classical hard spheres in infinitely many space dimensions are a notable exception. We show that, even without resorting to a Boltzmann distribution, dimensionality is a powerful organizing device for exploring the stationary properties of active hard spheres evolving far from equilibrium. In infinite dimensions, we exactly compute the stationary state properties that govern and characterize the collective behavior of active hard spheres: the structure factor and the equation of state for the pressure. In turn, this allows us to account for motility-induced phase separation. Finally, we determine the crowding density at which the effective propulsion of a particle vanishes.