Here's pretty decent deal on a health club membership: $299.99 (aka $300) at Sam's Club for a 2 -year membership at 24-Hour Fitness. That's $12.50/month for unlimited showers and hot tub soaks at the vast majority of 24-Hour Fitness clubs in the USA (excluding only "Super Sport" clubs in certain wealthy neighborhoods). Even if you had to join Sams Club ($35) just to take advantage of this offer, you'd still be getting a fantastic deal.
A health club membership like this means you could sleep in rough circumstances (like camping or in your car) and still maintain your personal hygiene. If you slept in a car, stored your clothes in a storage unit and showered at a health club, your real quality of life (and your appearance to others) isn't going to be different than living in a fixed residence. As long as you aren't into entertaining and don't need television, this is a sustainable lifestyle.
24-Hour Fitness is a national chain, but it isn't located everywhere. It is strongest on the West Coast, especially Southern California, Las Vegas and Phoenix. (Here's their location map.) This chain is not helpful in the Northeast, although there is one Sport club (covered by the above plan) in the Bronx, New York City. There are even clubs in Hawaii, on both Oahu and Maui. (Rent a car, sleep in it and shower at the club, and you can avoid rip-off hotel bills.) In my favorite camping city, San Diego, there are clubs everywhere, some of them adjacent to the trolley line. Even during the rains last February, when my tent got swamped, I could still walk to the 24-Hour Fitness and soak in the hot tub.
24-Hour Fitness seems to be the best club for this purpose, at least in the Southwest. The "24-hour" feature can be extremely helpful at times. Another chain, LA Fitness, has a stronger presence in the Northeast (location map), but I'm not familiar with their pricing or policies. Golds Gym has more locations than both clubs combined, but each club is individually owned and they may not have a national plan.
If you know of any other useful health club deals, let me know.