We characterize the star KOI 961, an M dwarf with transit signals indicative of three short-period exoplanets discovered by the Kepler mission. We proceed by comparing KOI 961 to Barnard's Star, a nearby, well-characterized mid-M dwarf. We compare colors, optical and near-infrared spectra, and find remarkable agreement between the two, implying similar effective temperatures and metallicities. Both are metal-poor compared to the Solar neighborhood, have low projected rotational velocity, high absolute radial velocity, large proper motion, and no quiescent H alpha emission-all of which are consistent with being old M dwarfs. We combine empirical measurements of Barnard's Star and expectations from evolutionary isochrones to estimate KOI 961's mass (0.13 + or - 0.05 M), radius (0.17 + or - 0.04 R), and luminosity (2.40 x 10 super(-3.0 + or - 0.3) L). We calculate KOI 961's distance (38.7 + or - 6.3 pc) and space motions, which, like Barnard's Star, are consistent with a high scale-height population in the Milky Way. We perform an independent multi-transit fit to the public Kepler light curve and significantly revise the transit parameters for the three planets. We calculate the false-positive probability for each planet candidate, and find a less than 1% chance that any one of the transiting signals is due to a background or hierarchical eclipsing binary, validating the planetary nature of the transits. The best-fitting radii for all three planets are less than 1 R, with KOI 961.03 being Mars-sized (R sub(P) = 0.57 + or - 0.18 R), and they represent some of the smallest exoplanets detected to date.