  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%.
