[Abridged] We present a search for fast optical transients (~0.5 hr-1 day) using repeated observations of the Pan-STARRS1 Medium-Deep Survey (PS1/MDS) fields. Our search takes advantage of the consecutive g/r-band observations (16.5 min in each filter), by requiring detections in both bands, with non-detections on preceding and subsequent nights. We identify 19 transients brighter than 22.5 AB mag (S/N>10). Of these, 11 events exhibit quiescent counterparts in the deep PS1/MDS templates that we identify as M4-M9 dwarfs. The remaining 8 transients exhibit a range of properties indicative of main-belt asteroids near the stationary point of their orbits. With identifications for all 19 transients, we place an upper limit of R_FOT(0.5hr)<0.12 deg^-2 d^-1 (95% confidence level) on the sky-projected rate of extragalactic fast transients at <22.5 mag, a factor of 30-50 times lower than previous limits; the limit for a timescale of ~day is R_FOT<2.4e-3 deg^-2 d^-1. To convert these sky-projected rates to volumetric rates, we explore the expected peak luminosities of fast optical transients powered by various mechanisms, and find that non-relativistic events are limited to M~-10 mag (M~-14 mag) for a timescale of ~0.5 hr (~day), while relativistic sources (e.g., GRBs, magnetar-powered transients) can reach much larger luminosities. The resulting volumetric rates are <13 (M~-10 mag), <0.05 (M~-14 mag) and <1e-6 Mpc^-3 yr^-1 (M~-24 mag), significantly above the nova, supernova, and GRB rates, respectively, indicating that much larger surveys are required to provide meaningful constraints. Motivated by the results of our search we discuss strategies for identifying fast optical transients in the LSST main survey, and reach the optimistic conclusion that the veil of foreground contaminants can be lifted with the survey data, without the need for expensive follow-up observations.
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