Clinical systems for optical coherence tomography (OCT) are used routinely to diagnose and monitor patients with a range of ocular diseases. They are large tabletop instruments operated by trained staff, and require mechanical stabilization of the head of the patient for positioning and motion reduction. Here we report the development and performance of a robot-mounted OCT scanner for the autonomous contactless imaging, at safe distances, of the eyes of freestanding individuals without the need for operator intervention or head stabilization. The scanner uses robotic positioning to align itself with the eye to be imaged, as well as optical active scanning to locate the pupil and to attenuate physiological eye motion. We show that the scanner enables the acquisition of OCT volumetric datasets, comparable in quality to those of clinical tabletop systems, that resolve key anatomic structures relevant for the management of common eye conditions. Robotic OCT scanners may enable the diagnosis and monitoring of patients with eye conditions in non-specialist clinics.