We show that exciton-polariton condensates may exhibit a fundamental, self-localized nonlinear excitation in quantum hydrodynamical systems, which takes the form of a dark ring-shaped breather. We predict that these structures form spontaneously and remain stable under a combination of uniform resonant and nonresonant forcing. We study single-ring dynamics, ring interactions, and ring turbulence, and explain how direct experimental observations might be made. We discuss the statistics of ring formation and propose an experimental scheme by which these structures may be exploited to study the smooth crossover between equilibrium and nonequilibrium critical phase transitions. The observation of a breathing-ring soliton would represent a fundamental breathing soliton within the broad field of quantum hydrodynamics.