you are starting down a long, long road.....
LONG.
If you really, really, really want to do it...
I would go for a used 8 track system, either casette or digital, hard disk. something with a mixer built in. your 8 tracks would be 2 vocals, 3 drums, 2 guitars, 1 bass. if you can run the bass direct, you need 7 mics for the above. herein lies your problem, 7 el cheapo mics with cables is about 700 bucks alone. then you have to think about a way to mix it down, maybe to your pc soundcard? plus you need monitors, which are like 300 minimum. and you need them. I used to use a stereo, I pretty much ruined my hearing and my mixes sucked. so you're looking at probably 2x what you want to spend. you could try to find a 'lot' on ebay of some cheap vintage mics, from a church or something...
you mentioned reel to reel, the tape also costs money so that would also contribute to going beyond your budget. but you could find a cheap mixer and mix that down to a reel to reel recorder, if you could get enough mics and cablesfor about $100...you maybe could stay in budget if you dont count the cost of tape.