  [Abridged] We present the first comprehensive analysis of HST observations of
short-duration gamma-ray burst (GRB) host galaxies. These observations allow us
to characterize the galactic and local environments of short GRBs as a powerful
constraint on the nature of their progenitors. Using the HST data for 10 short
GRB hosts we determine the host morphological properties, measure precise
physical and host-normalized offsets relative to the galaxy centers, and study
the locations of short GRBs relative to their host light distributions. We find
that most short GRB hosts have exponential disk profiles, characteristic of
late-type galaxies, but with a median size that is twice as large as that of
long GRB hosts, commensurate with their higher luminosities. The observed
distribution of projected physical offsets, supplemented by ground-based
measurements, has a median of ~5 kpc, about 5 times larger than for long GRBs,
and in good agreement with predicted offset distributions for NS-NS binary
mergers. For the short GRB population as a whole we find the following robust
constraints: (i) >25% have projected offsets of <10 kpc; and (ii) >5% have
projected offsets of >20 kpc. We find no clear systematic trends for the offset
distribution of short GRBs with and without extended soft emission. While the
physical offsets are larger than for long GRBs, the distribution of
host-normalized offsets is nearly identical due to the larger size of short GRB
hosts. Finally, unlike long GRBs, which are concentrated in the brightest
regions of their host galaxies, short GRBs appear to under-represent the light
distribution of their hosts; this is true even in comparison to core-collapse
and Type Ia supernovae. Based on these results, we conclude that short GRBs are
consistent with a progenitor population of NS-NS binaries.
