1. Budget backpacking. The secret appeal of backpacking is that once you've mastered it, theoretically you can go anywhere on Planet Earth--roads no longer necessary. As a hobby, it can eat up as much money as you want to pour into it, but it doesn't have to. Every Appalachian trail hiker knows the story of Grandma Gatewood, who hiked all 2,168 miles of the trail three times, at ages ranging from 67 to 75, equipped with little more than sneakers, an old Army blanket, and an old shower curtain.
You can spend hundreds of dollars on a backpacking tent, or you can sleep under a sheet of plastic costing at most a few dollars (that's mine pictured above). The latter will give you more space, and--if you know what you're doing--may well keep you drier. You can make your own backpack out of a mesh laundry bag and a old book bag for another handful of dollars (check out the "Sgt. Rock Rucksack". You can make your own alcohol stove out of three empty soda cans, eat cup noodles, carry (and refill) a store-bought liter of water rather than a canteen, and so on. At online forums, others are more than happy to help you get started.
2. Paper modeling. Also known as card modeling, or papercraft. Cut paper, glue it together, you get a ship, an airplane, a funny animal, or whatever your passion is. See some examples at the Currell.net Gallery. Basic resources needed: a computer and printer, some heavyweight paper, a hobby knife, some white glue. Difficulty ranges from slightly glorified paper airplanes to complicated patterns with hundreds of parts. Thousands of patterns are available on-line for free, others for sale. You can buy books with preprinted patterns.
Or go hard-core: just you, armed with a knife and glue, versus a blank stack of paper. If you want to go a little deeper you can design your own patterns (and perhaps share them on-line).
Here's some places to start looking, just a few out of a large universe: Currell.net, Lower Hudson Valley Gift Shop (despite the name, everything is free, although they do accept donations), Papercraft Paradise.
3. Parkour. Also known as free-running. If you've never seen this, it is difficult to explain. It's a sport with no rules, a game with no winners or losers. The objective is to get from point A to point B as gracefully and athletically as possible. You can find some in the movies--the big chase seen at the beginning of Casino Royale features parkour practitioner Sébastien Foucan as the bomb-maker. Or check out the following video:
Obviously parkour can be extremely athletic (I've never tried it--as yet), but even these guys had to start small.
4. Chess. Sixty-four squares, sixteen pieces, and after several hundred years the possibilities are still endless. You can score a set for a few dollars (I've picked up some at the dollar store), and there are plenty of resources on-line to teach you how to play and how to play better. If you have no friends, you can download free software to play against (and you may not need a board). You can find an adversary on-line at any time.
5. Dancing. In my misspent youth, I did the standard backpacking trip to Europe, during which a friend and I made a side trip to Morocco. During this time we stayed for three days with a family in Casablanca. This says something about variations in standards of hospitality around the world. We had never met them before but they insisted, notwithstanding there were seven of them living in a (clean and dignified) three-room apartment (and even though, as we later discovered, they had other house guests arriving in the middle of the night).
I mention the size of the apartment to emphasize that these were not people who had extra money to waste on frivolities. One of the neighborhood kids did have a hand-held cassette player, along with two cassettes--I recall one was Simon and Garfunkel and the other was the soundtrack to Grease. Dancing was an everyday activity, taken up at the spur of the moment. They danced with a natural grace that comes with a lot of practice. And they had fun.