We critically analyze the measurement of galaxy cluster gas masses, which is central to cosmological studies that rely on the galaxy cluster gas mass fraction. Using synthetic observations of numerically simulated clusters viewed through their X-ray emission and thermal Sunyaev-Zeldovich effect (SZE), we reduce the observations to obtain measurements of the cluster gas mass. We are thus able to quantify the possible sources of uncertainty and systematic bias associated with the common simplifying assumptions used in reducing real cluster observations including isothermality and hydrostatic equilibrium. We find that intrinsic variations in clusters limit the precision of observational gas mass estimation to ~10% to 1 sigma confidence excluding instrumental effects. Gas mass estimates performed via all methods surprisingly show little or no trending in the scatter as a function of cluster redshift. For the full cluster sample, methods that use SZE profiles out to roughly the virial radius are the simplest, most accurate, and unbiased way to estimate cluster mass. X-ray methods are systematically more precise mass estimators than are SZE methods if merger and cool core systems are removed, but X-ray methods slightly overestimate (5-10%) the cluster gas mass on average. We find that cool core clusters in our samples are particularly poor candidates for observational mass estimation, even when excluding emission from the core region. The effects of cooling in the cluster gas alter the radial profile of the X-ray and SZE surface brightness outside the cool core region, leading to poor gas mass estimates in cool core clusters. Finally, we find that methods using a universal temperature profile estimate cluster masses to higher precision than those assuming isothermality.