On thing my dad did to help me learn to judge the size of the car is this: we found a place with a set of speed bumps aligned next to each other (not in the direction of the road, but across the road).
I then practiced driving with one wheel going in the space between the two bumps. You can easily tell if you got it right by the car not rocking on that side. So that’s width.
For depth, you can just generally be conservative and get by OK. It’s useful to learn with a car whose nose you can easily see (that doesn’t mean it has to be long, it just has to not slope down too much).

# I often don’t get enough sleep to be able to drive safely.

Yes, well, that is a problem. Don’t drive sleepy; I’ve nodded off a couple times and been exceedingly lucky doing it.

# Owning a car is expensive. Even if I had enough money, it seems like there would be better ways to spend it.

Learning to drive does not imply owning a car. It implies mobility when you need it, when you decide its worthwhile.

# If I had a car, I wouldn’t walk to school/work, and then I would get even less exercise than I do now.

So don’t do that.