A luminous optical transient (OT) that appeared in NGC 300 in early 2008 had a maximum brightness, M_V ~ -12 to -13, intermediate between classical novae and supernovae. We present ground-based photometric and spectroscopic monitoring and adaptive-optics imaging of the OT, as well as pre- and post-outburst space-based imaging with HST and Spitzer. The optical spectrum at maximum showed an F-type supergiant photosphere with superposed emission lines of hydrogen, Ca II, and [Ca II], similar to the spectra of low-luminosity Type IIn supernova impostors like SN 2008S, as well as cool hypergiants like IRC +10420. The emission lines have a complex, double structure, indicating a bipolar outflow with velocities of ~75 km/s. The luminous energy released in the eruption was ~10^47 ergs, most of it emitted in the first 2 months. By registering new HST images with deep archival frames, we have precisely located the OT site, and find no detectable optical progenitor brighter than broad-band V magnitude 28.5. However, archival Spitzer images reveal a bright, non-variable mid-IR pre-outburst source. We conclude that the NGC 300 OT was a heavily dust-enshrouded luminous star, of ~10-15 Msun, which experienced an eruption that cleared the surrounding dust and initiated a bipolar wind. The progenitor was likely an OH/IR source which had begun to evolve on a blue loop toward higher temperatures, but the precise cause of the outburst remains uncertain.