Prior to the launch of NuSTAR, it was not feasible to spatially resolve the hard (E > 10 keV) emission from galaxies beyond the Local Group. The combined NuSTAR dataset, comprised of three ~165 ks observations, allows spatial characterization of the hard X-ray emission in the galaxy NGC 253 for the first time. As a follow up to our initial study of its nuclear region, we present the first results concerning the full galaxy from simultaneous NuSTAR, Chandra, and VLBA monitoring of the local starburst galaxy NGC 253. Above ~10 keV, nearly all the emission is concentrated within 100 of the galactic center, produced almost exclusively by three nuclear sources, an off-nuclear ultraluminous X-ray source (ULX), and a pulsar candidate that we identify for the first time in these observations. We detect 21 distinct sources in energy bands up to 25 keV, mostly consisting of intermediate state black hole X-ray binaries. The global X-ray emission of the galaxy - dominated by the off-nuclear ULX and nuclear sources, which are also likely ULXs - falls steeply (photon index >~ 3) above 10 keV, consistent with other NuSTAR-observed ULXs, and no significant excess above the background is detected at E > 40 keV. We report upper limits on diffuse inverse Compton emission for a range of spatial models. For the most extended morphologies considered, these hard X-ray constraints disfavor a dominant inverse Compton component to explain the {gamma}-ray emission detected with Fermi and H.E.S.S. If NGC 253 is typical of starburst galaxies at higher redshift, their contribution to the E > 10 keV cosmic X-ray background is < 1%.
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