An illustrative example embodiment of an elevator system includes a platform and a plurality of supports configured to selectively engage a nearby structure. The plurality of supports include at least a first support and a second support. The first and second supports alternate between engaging the nearby structure to support the platform while the other one of the supports is disengaged from the nearby structure. At least one of the first and second supports is configured to move relative to the platform while engaging the nearby structure to cause vertical movement of the platform.