High mobility, strong spin-orbit coupling, and large Lande g-factor make the two-dimensional electron gas (2DEG) in InAs quantum wells grown on nearly-lattice-matched GaSb substrates an attractive platform for mesoscopic quantum transport experiments. Successful operation of mesoscopic devices relies on three key properties: electrical isolation from the substrate; ability to fully deplete the 2DEG and control residual sidewall conduction with lithographic gates; and high mobility to ensure ballistic transport over mesoscopic length scales. Simultaneous demonstration of these properties has remained elusive for InAs 2DEGs on GaSb substrates. Here we report on heterostructure design, molecular beam epitaxy growth, and device fabrication that result in high carrier mobility and full 2DEG depletion with minimal residual edge conduction. Our results provide a pathway to fully-controlled 2DEG-based InAs mesoscopic devices.