Gabriel, as someone pointed out, a "hot rod" can be anything -- even a Dodge Neon or a Honda Civic. If you want a TRADITIONAL rod, well, 30's cars aren't considered junk that nobody wants anymore, as they were as late as the early 70s. The mid 70s is about when collector car interest really started picking up on everything. In the 50s and 60s someone interested in preserving/restoring old cars were considered a bit on the odd/eccentric side, and those who modified them for driving, even when improving performance, either really odd or to cheap to buy new. Those were the days when it was common to at least get a newer used car every 2-5 years.
Sounds like you don't know much about working on cars right now. So what I'd recommend as the best course of action is to skip the "rat rod" craze for a few years. They will still be around then, and most of the fad craze might be over, leaving a lot of cars for the "true believers" (kind of like what happened to the Pro-Street craze after the mid 90s).
Find an older car you like, then get a Haynes or Chilton's manual for it. Haynes seems to be the best for the newbie -- they quickly tell you what's best handled by a pro and you should leave it alone, but if there's a relatively easy test procedure they'll ooutline it for you. Don't worry whether it's EFI or cabred. You'll learn a lot with either. An older model Japanese car is just as good as an American car. you might want to use Google and see how many speed parts are made for it before buying though, or how many sites outline mods you're interested in.
Don't overlook "underappreciated" cars -- 50s and 60s Ramblers, Studebakers, even some Chevys and Fords. 46-54 Chevys are still out there reasonably priced, but fords in those eras are more popular. Remember -- more popular = MORE EXPENSIVE for initial purchase. Might be more parts available, but you're talking about hot rodding -- part of that is making things work that weren't necessarilly intended for your vehicle in a SAFE manner. That takes using your head and developing skills, not necessarily spending a lot of money. There are "REAL" hot rodders and there are "CHECKBOOK RODDERS" (or "credit card rodders", take your pick!). Which do you want to be?
A "classic" example of using an under appreciated car is my 63 Rambler Classic wagon (see my photo album). Flamed four-door wagon, 4.6L fuel injected in-line six, manually controlled electronic auto trans, Jaguar rear axle, Ford T-bird rack and pinion steering, Ford Ranger brake booster and master cylinder, Eagle Premier six way power front seats, rear seat, and console, factory AC unit in the dash coupled to a modern compressor and condensor, and lots of other little mods. All the above required some amount of fabrication and ingenuity to make work properly and safely. I found 80% of the info I needed on the Internet. You can find a lot more now!
There's nothing wrong with four doors, especially for a first learning car. Fewer people want them, so they are much cheaper (except in rare cases like the 56-57 Chevy -- two doors are so expensive that people are settling for four doors, driving demand for them up too). It may not be as cool looking as a two door, but it's a lot easier to take friends with you! I had a little four door wagon in high school (graduated in 1980!). Very NOT COOL back then to most, but I liked it anyway! The cool part was if more than two of us wanted to go anywhere everyone preferred to go in my car so they didn't have to climb in and out of the back seat. If it was cold out and we were sitting in the school parking lot, everyone piled in my car to sit a while because it was easier. So it can be cool to have a "more-door".