In fact, a lot of colleges have a pool of money for just this kind of activity.
Rutgers used some of this money to organize a paintball trip. On the way back from that trip is how the club got started which then turned into a team etc. I was one of the first officers and that's what led to me working in pro paintball several years later.
Just doing the napkin math on it, it doesn't come out to be a crazy number, and the bang per buck (in terms of how much it benefits the students) was great.
Another similar event that my school did was reserving GA Aquarium for students for an evening+overnight. That one was even better imo, as I love GA Aquarium a ton. And being allowed to stay overnight to sleep in your own sleeping bag right next to sharks and other animals passing you by was soothing and awesome.