Hiking. I am also terrible at any sport involving a high-speed ball. If you want some camaraderie, join the Sierra Club and go on group hiking trips. They have them scheduled all the time all over the place. While you don't get to compete, you will get to experience nature and meet some very nice people. Depending on your area, they might even have urban conditioning walks where you walk through the city. (Which reminds me, I've been very bad about scheduling in some Sierra Club time...)
Orienteering. I actually just found about this yesterday. It's a competitive sport called Orienteering, which involves trekking through a space, like the woods, and finding your way with a compass and a map. In some cases, participants are timed to see how quickly they can return from the woods.
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