We use cosmological gasdynamic simulations to investigate the accuracy of galaxy cluster mass estimates based on X-ray observations. The experiments follow the formation of clusters in different cosmological models and include the effects of gravity, pressure gradients, and hydrodynamical shocks. A subset of our ensemble also allows for feedback of mass and energy from galactic winds into the intracluster medium. We find that mass estimates based on the hydrostatic, isothermal β- model are remarkably accurate when evaluated at radii where the cluster mean density is between 500 and 2500 times the critical density. At lower densities, radial temperature information becomes important. In the quoted radial regime, the distribution of the estimated-to-true mass ratio, derived from 174 artificial images constructed from the simulations, is nearly unbiased and has a standard deviation of 14%-29%. The scatter can be considerably reduced (to 8%-15%) by using an alternative mass estimator that exploits the tightness of the mass- temperature relation found in the simulations. The improvement over β-model estimates is due to the elimination of the variance contributed by the gas outer slope parameter. We discuss these findings and their implications for recent measurements of cluster baryon fractions.