A technique is proposed for producing high average intensity X-ray radiation from a storage ring for studies of the ultrafast phenomena on a subpicosecond time scale. Two RF cavity accelerating structures excited in the E 110 mode can be installed in a storage ring to create vertical displacements of electrons correlated with their longitudinal position in the bunch. The magnitude of these displacements can be sufficient for the X-ray radiation of the electron bunch between accelerating structures to be viewed as produced by a large number of independent sources, each of a subpicosecond duration.