The fastback in my signature was originally restored when I was 19 and in college! It can be done. I think my budget at the time was about $3000 or so, after borrowing the cost of the car from my parents. My dad did the welding for me on the minor sheetmetal replacement we did, battery apron, taillight panel, etc. The quarters needed replaced but we smoothed them with bondo knowing the goal was to get me through college and then I'd do them again later when I had more cash. A friend of my dad's did paint work on the side and painted my car for the cost of the paint. BTW I was only making about $6 an hour in college myself at the time. I did stuff like reuse the original control arms and just replaced the ball joints. I rebuilt the original 4-wheel drum brakes. Engine was machined and we rebuilt it at home ourselves.
This is counter to the mindset of doing it right and do it once, as the reality is you don't always have the expense for that. Just realize if you do some of that, you end up spending more money. Example, if you replace ball joints now to save the expense of a completely new control arm, eventually when you do buy a new control arm you are out the cost of the ball joint. If you bondo the quarter panels and then go back later, you are out the cost of the bondo and the first paint job because it all gets done again. If you know that going in, and are ok with that, you can build a very nice driver and have fun with it on a pretty tight budget. My car was always a "20 footer" due to the quality of the paint work, etc, but I had a blast with it, it got a lot of attention, and I ended up putting 100,000 miles on it before I tore it down again to "do it right."
BTW, as for the old age thing, I bought my car when I was 18, started driving it when I was 19 (got it to school over spring break sophomore year), and then tore it apart again when I was 30. I'm now 35 and hope to be driving it regularly any time now.