We use umbrella sampling Monte Carlo and forward and reverse forward flux sampling (FFS) simulation techniques to compute the free energy barriers to evaporation of water confined between two hydrophobic surfaces separated by nanoscopic gaps, as a function of the gap width, at 1 bar and 298 K. The evaporation mechanism for small (1 X 1 nm(2)) surfaces is found to be fundamentally different from that for large (3 X 3 nm(2)) surfaces. In the latter case, the evaporation proceeds via the formation of a gap-spanning tubular cavity. The 1 x 1 nm(2) surfaces, in contrast, are too small to accommodate a stable vapor cavity. Accordingly, the associated free energy barriers correspond to the formation of a critical-sized cavity for sufficiently large confining surfaces, and to complete emptying of the gap region for small confining surfaces. The free energy barriers to evaporation were found to be of O(20kT) for 14 angstrom gaps, and to increase by approximately similar to 5kT with every 1 angstrom increase in the gap width. The entropy contribution to the free energy of evaporation was found to be independent of the gap width.