I was at McMurdo Station for three weeks last October. There are no skating rinks, and I've never heard of anyone going ice skating.
That doesn't mean it is impossible, as I saw some smooth ice in a few places, but ice skating depends on the metal skates melting a very thin layer of surface ice in order to glide on it. At McMurdo, it would likely be too cold most of the year, and then too warm for much of the summer. If you did want to try ice skating you would probably have only a short window of opportunity.
The two places I saw smooth ice were on the ocean (the sea-ice). Unfortunately, by the time it is warm enough to skate, it may be too warm to safely go on the sea-ice. The other place was on Hut Ridge. When I went on the Hut Ridge Trail, I saw several spots where the ice and snow had melted the previous summer and had frozen solid during the winter - fairly smooth. There might be a short time when that spot (fairly small) could be used for skating.
Other facilities, as already noted, include a couple gymnasiums, a climbing wall, a bowling alley, walking, running, and skiing. If you brought a bicycle, that could be done as well. And of course there are some hills to climb.
This past year they asked for volunteers to assist some of the divers, and if you have experience and helped out a lot, they might
let you do a little diving - but the water is very cold!
Best of luck!