I started off with the Lord of the Rings game back when it first game out and I was only about 15-16. It was a relatively expensive hobby compared to some, but not bank-breakingly so. Obviously I couldn't (and still can't) just buy everything I might possibly ever want, but with some judicious list-writing and reasonable expectations of what your army will be, it's doable.
Prices have gone up since then, and Lord of the Rings was always a bit cheaper than 40k of WHFB. But I got into 40k when I was about 18 or 19, and managed to get a 1000 point Eldar army without much trouble. Over the next few years I slowly expanded, adding a tank here, some new Aspect warriors there. The up-front cost for the core of your army is always the biggest price-sink, but starting with a solid core of about 1000 points and then adding a squad at a time isn't horrendously expensive.
This idea that you need to play huge games or you're not doing it right is just nonsense. That attitude is mainly fuelled by vets who've been collecting for decades and just want to show off their collections. I've always found smaller games more strategically interesting, since every model counts for so much more.
Finally, for the very wallet-conscious, there's the skirmish games like Armageddon and Necromunda (and hopefully Mordheim again someday). Ages ago when I used to work at GW as a redshirt, I'd often recommend Necromunda or Mordheim to people who were curious about dipping their toes in the hobby, but didn't want to spend several hundred on an army up-front. Once box of Empire Militia or Catachans is enough to get a decent warband/gang put together, and then you could download the Specialist Games rulebooks for free. With Armageddon you've got to add the rulebook cost, but pick almost any box of troops from any army and you've got a kill team to start playing with.