Nanoscale size effects drastically alter the fundamental properties of semiconductors. Here, we investigate the dominant role of quantum confinement in the field-effect device properties of free-standing InAs nanomembranes with varied thicknesses of 5-50 nm. First, optical absorption studies are performed by transferring InAs "quantum membranes" (QMs) onto transparent substrates, from which the quantized sub-bands are directly visualized. These sub-bands determine the contact resistance of the system with the experimental values consistent with the expected number of quantum transport modes available for a given thickness. Finally, the effective electron mobility of InAs QMs is shown to exhibit anomalous field and thickness dependences that are in distinct contrast to the conventional MOSFET models, arising from the strong quantum confinement of carriers. The results provide an important advance toward establishing the fundamental device physics of two-dimensional semiconductors.