Exoskeletons for daily use by those with mobility impairments are being developed. They will require accurate and robust scene understanding systems. Current research has used vision to identify immediate terrain and geometric obstacles, however these approaches are constrained to detections directly in front of the user and are limited to classifying a finite range of terrain types (e.g., stairs, ramps and level-ground). This paper presents Exosense, a vision-centric scene understanding system which is capable of generating rich, globally-consistent elevation maps, incorporating both semantic and terrain traversability information. It features an elastic Atlas mapping framework associated with a visual SLAM pose graph, embedded with open-vocabulary room labels from a Vision-Language Model (VLM). The device's design includes a wide field-of-view (FoV) fisheye multi-camera system to mitigate the challenges introduced by the exoskeleton walking pattern. We demonstrate the system's robustness to the challenges of typical periodic walking gaits, and its ability to construct accurate semantically-rich maps in indoor settings. Additionally, we showcase its potential for motion planning -- providing a step towards safe navigation for exoskeletons.