We present an analysis of the photometry and spectroscopy of the host galaxy of Swift-detected GRB 080517. From our optical spectroscopy, we identify a redshift of z = 0.089 plus or minus 0.003, based on strong emission lines, making this a rare example of a very local, low-luminosity, long gamma-ray burst. The galaxy is detected in the radio with a flux density of S(4.8 GHz) = 0.22 plus or minus 0.04 mJy -- one of relatively few known gamma-ray bursts hosts with a securely measured radio flux. Both optical emission lines and a strong detection at 22 mu m suggest that the host galaxy is forming stars rapidly, with an inferred star formation rate similar to 16 Msun/yr and a high dust obscuration (E(B - V) > 1, based on sightlines to the nebular emission regions). The presence of a companion galaxy within a projected distance of 25 kpc, and almost identical in redshift, suggests that star formation may have been triggered by galaxy-galaxy interaction. However, fitting of the remarkably flat spectral energy distribution from the ultraviolet through to the infrared suggests that an older, 500 Myr post-starburst stellar population is present along with the ongoing star formation. We conclude that the host galaxy of GRB 080517 is a valuable addition to the still very small sample of well-studied local gamma-ray burst hosts.