5 miles is not a long way. I live in a metro area, one of the highest traffic-wise, and I go that far, no problem.
The car - you have to do this thing called "work" and work at finding someone who has a car and share. Maybe a church has a van that does this kind of thing. I'm not saying it is the easiest thing in the world, but seriously, it is not that difficult to find out if someone has a car that can go to a grocery store, even a normal one, and carpool.
There's a lot of other things one can do to lift themselves out of poverty, that are not difficult.