We present results from three nearly simultaneous Nuclear Spectroscopic Telescope Array ( NuSTAR ) and Chandra monitoring observations between 2012 September 2 and 2012 November 16 of the local star-forming galaxy NGC 253. The 3-40 keV intensity of the inner ~ 20 arcsec ( ~ 400 pc) nuclear region, as measured by NuSTAR , varied by a factor of ~ 2 across the three monitoring observations. The Chandra data reveal that the nuclear region contains three bright X-ray sources, including a luminous ( L2-10 keV ~ few × 1039 erg s-1 ) point source located ~ 1 arcsec from the dynamical center of the galaxy (within the 3σ positional uncertainty of the dynamical center); this source drives the overall variability of the nuclear region at energies ≳3 keV. We make use of the variability to measure the spectra of this single hard X-ray source when it was in bright states. The spectra are well described by an absorbed (NH ≈ 1.6 × 1023 cm-2) broken power-law model with spectral slopes and break energies that are typical of ultraluminous X-ray sources (ULXs), but not active galactic nuclei (AGNs). A previous Chandra observation in 2003 showed a hard X-ray point source of similar luminosity to the 2012 source that was also near the dynamical center (θ ≈ 0.4 arcsec); however, this source was offset from the 2012 source position by ≈ 1 arcsec. We show that the probability of the 2003 and 2012 hard X-ray sources being unrelated is ≫99.99% based on the Chandra spatial localizations. Interestingly, the Chandra spectrum of the 2003 source (3-8 keV) is shallower in slope than that of the 2012 hard X-ray source. Its proximity to the dynamical center and harder Chandra spectrum indicate that the 2003 source is a better AGN candidate than any of the sources detected in our 2012 campaign; however, we were unable to rule out a ULX nature for this source. Future NuSTAR and Chandra monitoring would be well equipped to break the degeneracy between the AGN and ULX nature of the 2003 source, if again caught in a high state.