Service brat, here, living in a big service town. (San Antonio, mother-in-law of the Army, home of basic training for the Air Force) and I am shocked that one poster here has been in the Air Force for a year and found no one to play with. The German AF must be very different socially than the American!
I've been playing with service people since I started to play in 1979 on a college campus. Since the game was a regular one - we had a room in the Student Union Building reserved for Sundays - we got players from nearby Ft. Sam dropping in. One of my friends is a jewelry maker, a large chunk of whose business has always been concessions at bases and posts; he used to live down the block from the Security Hill gate at Kelly AFB (since decommissioned) and would pick up players to bring home while chatting up customers. Since his most lucrative concessions were associated with training sites a lot of these were short-term and crossed a lot of services, but we also had long-term regulars contacted in this manner, including a training instructor whose freedom to play was tied to the basic training schedule. When my husband lived in the dorms at Kelly, I used to attend games in the day room with him.
Another pastime I've always seen a lot of involvement in was medieval recreation, specifically the Society for Creative Anachronism. I've mostly played tabletop with noncoms and enlisted men, but a lot of my SCA acquaintance was commissioned. This may be because the fraternization problems are easier to cope with in large outdoor gatherings that include formal social structures and frequently draw people from long distances. Also, whereas you might have to build a new gaming group from scratch every time you were reassigned or went for training, you could walk into a new town, look up the nearest regional SCA organization, walk in, and be with family.