Self-balancing exoskeletons are a key enabling technology for individuals with mobility impairments. While the current challenges focus on human-compliant hardware and control, unlocking their use for daily activities requires a scene perception system. In this work, we present Exosense, a vision-centric scene understanding system for self-balancing exoskeletons. We introduce a multi-sensor visual-inertial mapping device as well as a navigation stack for state estimation, terrain mapping and long-term operation. We tested Exosense attached to both a human leg and Wandercraft's Personal Exoskeleton in real-world indoor scenarios. This enabled us to test the system during typical periodic walking gaits, as well as future uses in multi-story environments. We demonstrate that Exosense can achieve an odometry drift of about 4 cm per meter traveled, and construct terrain maps under 1 cm average reconstruction error. It can also work in a visual localization mode in a previously mapped environment, providing a step towards long-term operation of exoskeletons.