A novel method for measuring nuclear reactions in inverse kinematics with stored ion beams was successfully used to extract the nuclear-matter radius of Ni-58. The experiment was performed at the experimental heavy-ion storage ring at the GSI facility using a stored Ni-58 beam at energies of 100 and 150 MeV/u and an internal helium gas-jet target. Elastically scattered a-recoils at low momentum transfers were measured with an in-ring detector system compatible with ultrahigh vacuum. Experimental angular distributions were fitted using density-dependent optical model potentials within the eikonal approximation. This permitted the extraction of the point-matter root-mean-square radius of Ni-58 with an average value of 3.70(7) fm. Results from this work are in good agreement with several experiments performed in the past in normal kinematics. This pioneering experiment demonstrates a major breakthrough towards future investigations with far-from-stability stored beams using the present technique.