A realistic EOS (equation of state) leads to strange stars (ReSS) which are compact in the mass-radius (M-R) plot, close to the Schwarzschild limiting line. Many of the observed stars fit in with this kind of compactness, irrespective of whether they are X-ray pulsars, bursters or soft γ repeaters or even radio pulsars. We point out that a change in the radius of a star can be small or large, when its mass is increasing and this depends on the position of a particular star on the M-R curve. We carry out a stability analysis against radial oscillations and compare with the EOS of other SS models. We find that the ReSS is stable and an M-R region can be identified to that effect.