As astronauts moved around outside the shuttle and space station, they always were tethered to the spacecraft to avoid floating away. Ring brackets like this were installed on handrails and other structural supports at extravehicular worksites. Astronauts clipped their safety tether hook into a ring and then had both hands free to do their task tasks. They also clipped tool tethers to these rings to keep tools secure. These tether rings came from the Hubble Space Telescope servicing equipment inventory.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.