We characterize the kinematics, morphology, stellar populations and star formation histories of a sample of massive compact galaxies in the nearby Universe, which might provide a closer look at the nature of their high-redshift (z >rsim 1.0) massive counterparts. We find that nearby compact massive objects show elongated morphologies and are fast rotators. New high-quality long-slit spectra show that they have young mean luminosity-weighted ages (2 Gyr) and metallicities solar or above ([Z/H] >rsim 0.0). No significant stellar population gradients are found. The analysis of their star formation histories suggests that these objects have experienced recently enormous bursts which, in some cases, represent unprecedented large fractions of their total stellar mass. These galaxies seem to be truly unique, as they do not follow the characteristic kinematical and stellar population patterns of present-day massive ellipticals, spirals or even dwarfs.