Starting with the first detection of an afterglow from a short-duration hard-spectrum gamma-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 (>~ 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 arcsec 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 tau = 1.3 +- 0.2 Gyr and the minimum kick velocity of the SHB progenitor is v_kick = 55 +- 15 km/s.
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