I always understood it as a perk of being within the service, and also an actual real facility used to keep service people fit (fighting fit, fit for service life etc). Without others getting in the way, holding up others in the queue.
I agree with Hangarshuffle, at 3 of the bases in my last 12 years, the service personnel often couldn't get in the gym for all the contractors and contractors' mates hogging the kit. Same with the station outdoor pitches, service guys couldn't get a tennis court or an astroturf pitch because they were constantly in use by local civilian clubs.
Veterans should be looked after, but Military gyms aren't a retiree perk or a service to recognize veterans, they're primarily a facility to keep the serving base personnel fit. Any "add on" memberships after that shouldn't come at the expense of the guys and girls who live there, but sadly that's what often happens.