We report Chandra observations of the Scd galaxy NGC 2139, which is known to host a recently formed (10 7.6 yr) nuclear star cluster. The star cluster is undetected in X-rays, with an upper bound on 0.5-7 keV luminosity of L X < 7.1 × 10 37 erg s −1 . This bound implies a bolometric accretion luminosity <0.3% of the Eddington luminosity for a black hole with the mass (∼3400 M ) expected from extrapolation of the M − σ relation. The lack of X-ray emission indicates that a black hole, if present, is not undergoing significant accretion at the current time. While the central cluster is undetected, the data reveal a substantial population of bright X-ray point sources elsewhere in this galaxy, with eight qualifying as ultraluminous X-ray sources with L X > 10 39 erg s −1 . We use archival Hubble Space Telescope images to identify candidate optical counterparts for seven Chandra sources, which in most cases have optical luminosities and spatial profiles consistent with star clusters. Compared with other galaxies, the number of luminous X-ray sources in NGC 2139 is larger by a factor of ∼4-10 than expected based on its present star formation rate and stellar mass. This finding can be understood if NGC 2139 has concluded a burst of star formation in the recent past, and suggests that this galaxy could be important for testing the use of X-ray source populations as a chronometer of star formation history.