We combine new high resolution imaging and spectroscopy from Keck/NIRC2, Discovery Channel Telescope/ DSSI, and Keck/HIRES with published astrometry and radial velocities to measure individual masses and orbital elements of the GJ 3305 AB system, a young (~20 Myr) M+M binary (unresolved spectral type M0) member of the beta Pictoris moving group comoving with the imaged exoplanet host 51 Eri. We measure a total system mass of 1.11 + or - 0.04 M sub(middot in circle), a period of 29.03 + or - 0.50 year, a semimajor axis of 9.78 + or - 0.14 AU, and an eccentricity of 0.19 + or -0.02. The primary component has a dynamical mass of 0.67 + or - 0.05 M sub(middot in circle) and the secondary has a mass of 0.44 + or -0.05 M sub(middot in circle). The recently updated BHAC15 models are consistent with the masses of both stars to within 1.5sigma. Given the observed masses the models predict an age of the GJ 3305 AB system of 37 + or - 9 Myr. Based on the observed system architecture and our dynamical mass measurement, it is unlikely that the orbit of 51 Eri b has been significantly altered by the Kozai-Lidov mechanism.