Transition metal carbides include a wide variety of materials with attractive properties that are suitable for numerous and diverse applications. A most recent experimental advance could provide a path toward the successful synthesis of large-area and high-quality ultrathin Mo2C membranes with superconducting properties. In the present study, we used first-principles density functional theory calculations to explore the mechanical and optical response of single-layer and free-standing Mo2C. Uniaxial tensile simulations along the armchair and zigzag directions were conducted and we found that while the elastic properties are close along various loading directions, the nonlinear regimes in stress-strain curves are considerably different. We found that Mo2C sheets present negative Poisson's ratio and thus can be categorized as an auxetic material. Our simulations also reveal that Mo2C films retain their metallic electronic characteristic upon uniaxial loading. We found that for Mo2C nanomembranes the dielectric function becomes anisotropic along in-plane and out-of-plane directions. Our findings can be useful for the practical application of Mo2C sheets in nanodevices.