The control of molecules is key to the investigation of quantum phases, in which rich degrees of freedom can be used to encode information and strong interactions can be precisely tuned(1). Inelastic losses in molecular collisions(2-5), however, have greatly hampered the engineering of low-entropy molecular systems(6). So far, the only quantum degenerate gas of molecules has been created via association of two highly degenerate atomic gases(7,8). Here we use an external electric field along with optical lattice confinement to create a two-dimensional Fermi gas of spin-polarized potassium-rubidium (KRb) polar molecules, in which elastic, tunable dipolar interactions dominate over all inelastic processes. Direct thermalization among the molecules in the trap leads to efficient dipolar evaporative cooling, yielding a rapid increase in phase-space density. At the onset of quantum degeneracy, we observe the effects of Fermi statistics on the thermodynamics of the molecular gas. These results demonstrate a general strategy for achieving quantum degeneracy in dipolar molecular gases in which strong, long-range and anisotropic dipolar interactions can drive the emergence of exotic many-body phases, such as interlayer pairing and p-wave superfluidity.