Starting with the first detection of an afterglow from a short-duration hard-spectrum γ-ray burst (SHB) by Swift last year, a growing body of evidence has suggested that SHBs are associated with an older and lower redshift galactic population than long-soft GRBs and, in a few cases, with large (>åisebox-0.5ex 10 kpc) projected offsets from the centers of their putative host galaxies. Here we present observations of the field of GRB 060502B, a SHB detected by Swift and localized by the X-Ray Telescope (XRT). We find a massive red galaxy at a redshift of z=0.287 at an angular distance of 17.1$^″$ from our revised XRT position. Using associative and probabilistic arguments, we suggest that this galaxy hosted the progenitor of GRB 060502B. If true, this offset would correspond to a physical displacement of 73+/-19 kpc in projection, about twice the largest offset inferred for any SHB to date and almost an order of magnitude larger than a typical long-soft burst offset. Spectra and modeling of the star formation history of this possible host show it to have undergone a large ancient starburst. If the progenitor of GRB 060502B was formed in this starburst episode, the time of the GRB explosion since birth is τi̊sebox-0.5ex 1.3+/-0.2 Gyr and the minimum kick velocity of the SHB progenitor is v$_kick,min$=55+/-15 km s$^-1$.