Bowling is kind of a clutch sport for a Federation starship, if you think about it.
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Bowling alleys are long and narrow, which means you can fit one in by shaving a little off a lot of rooms instead of a lot off a few rooms. They're infinitely scalable, too; you can have a 2-lane alley, or a 200-lane alley, whatever fills your space. You can even stack them; if you've got two three-lane-wide spaces on adjacent decks, just add a staircase.
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Bowling accomodates a wide range of player counts; a lane might be one guy whose friends are all on-shift, trying to nail a perfect game, or it might be a scheduled five-member team competing against four other teams. There's not a lot of "big room" sports that take up less space per player than bowling. Especially in the future, where the machinery is probably one micro-repulsor emitter in the ball return and a tiny antigrav in each pin.
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Bowling is easy to teach: even aliens who've never heard of it can pretty easily pick up "roll this ball across that floor to hit those sticks".
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Somewhat relatedly, bowling is very social; if you're on a team, you're not actually playing n-1 shares of the time, so you can chat with your teammates, other players, etc. Good for both crew bonding, and for diplomacy.