While entropy is understood to dictate the elastic mechanics of elastomers, a consensus has not been reached regarding why they stiffen when confined to thin films or even the magnitude of this effect. Here, we present a combined computational and experimental approach to measure the confinement-induced stiffening of thin elastomer films using nanoindentation. First, we use finite element computations to modify the adhesion-free contact model to include the presence of a rigid substrate and verify this correction using indentation experiments on macroscopic polydimethylsiloxane (PDMS) films. Next, we validate this correction factor in indentation experiments where adhesion plays a major role through a series of experiments on micrometer-scale PDMS films. With this correction factor in hand, we perform an extensive series of nanoindentation experiments on submicrometer thickness films of varying cross-link density using a variety of probe radii and probe surface energies. We observe a consistent stiffening of all PDMS thin films that, interestingly, converges to a consistent elastic modulus at thicknesses similar to 100 nm regardless of the cross-link density. To explain these results, we propose a surface cross-linking model where the density of crosslinks between chains increases near the surface of the film because of oxidation-induced bonds between chains. Importantly, the apparent moduli predicted by this entropic model are in good agreement with all experimental results. Collectively, these results shed new light on the nanomechanics of elastomers and provide a general method for predicting the size-dependent effective moduli of soft polymers.