Electron storage rings used for the production of synchrotron radiation (SR) have an output photon brightness that is limited by the equilibrium beam emittance. By using interleaved injection and ejection of bunches from a source with repetition rate greater than 1 kHz, we show that it is practicable to overcome this limit in rings of energy ∼1 GeV. Sufficiently short kicker pulse lengths enable effective currents of many milliamperes, which can deliver a significant flux of diffraction-limited soft x-ray photons. Thus, either existing SR facilities may be adapted for nonequilibrium operation, or the technique applied to construct SR rings smaller than their storage ring equivalent.