We present the first sub-arcsecond resolution two-dimensional stellar kinematics and X-ray observations of the prototypical starburst galaxy NGC 253 which define the position and nature of the galactic nucleus. These observations comprise some of the best probes of the central 300 pc of NGC 253, the nearest massive galaxy undergoing a powerful starburst, and will allow us to gain more insight into the nature of the centers of starburst galaxies. We get an estimate of the stellar kinematic center location corresponding to an area of r ~ 1farcs2 centered ~0farcs7 southwest (SW) from the radio core, and historically presumed nucleus, TH2. Newly processed Chandra data reveal a central point-like hard X-ray source (X-1) lying ~0farcs4 SW from the kinematic center. Very accurate alignment between radio, infrared, and X-ray sources in the nuclear region shows that TH2, the IR photometric center, and X-1 are not associated with each other. As the kinematic center is consistent with the positions of TH2 and X-1, and both could be a manifestation of nuclear activity, we consider the two as possible galactic nucleus candidates. Although TH2 is the strongest compact radio source in the nuclear region, it does not have any infrared, optical, or X-ray counterparts. If the kinematic center is associated with this source, by analogy we suggest that the nucleus of NGC 253 resembles our Galactic center Sgr A*. On the other hand, X-1 is a heavily absorbed object (NH = 7.5 × 1023 cm-2) only detected at energies >2 keV (L 2-10 keV ~ 1040 erg s-1). If X-1 is instead associated with the kinematic center, the nucleus of NGC 253 is compatible with an obscured low-luminosity active galactic nucleus (AGN) or a spatially resolved super star cluster (SSC) brightening up in X-rays most probably due to young supernovae or supernova remnants, a situation also observed in the nuclear starburst of M82. If no SSC is associated with the kinematic center, we conclude that NGC 253 is a galaxy in which a strong starburst and a weak AGN (either TH2 or X-1) coexist. Results from few other high-resolution studies of nearby starburst galaxies (e.g., M82, NGC 5253, NGC 4945) indicate that the AGN in these systems, if present, is always in the low-luminosity regime. This may indicate that the onset of nuclear activity in galaxies is closely related with the occurrence of star formation, and that we are witnessing the emergence or disappearance of an AGN. Based on observations at the European Southern Observatory VLT (075.A-0250).