Can you put him in his crib with a darkened room (room darkening blinds aren't too expensive if you shop around) - or you can add room-darkening lining (available at fabric stores) to existing shades/curtains. My son enjoyed a white noise machine and some soft lullabies to begin with. The white noise machine kept out the street/house noises. Tell him it's quiet time, put him in his crib, and maybe put a few soft toys in there. Maybe he will fall asleep. But he definitely needs the down time - so do you! - so resist the urge to go get him. You are right that you can't depend on nursing or car rides - once you stop and try to put him down, he wakes up. Probably he needs to practice self-calming. At preschools/day care, the kids all have to lie down on a mat - they don't require them to sleep, but it's definitely quiet time. Maybe they get a book but they have to stay on their mats. 90% of them fall asleep.
My son napped for 3 hours well past his 4th birthday - he was very active in the morning and he definitely needed it. In fact, I didn't start him in kindergarten at age 5 because it would have been afternoon session - there's no way he would have been able to handle it because he really had only given up the nap 6 months before. Holding him out until 6 (and then a morning session) was the best decision we ever made. Anything you can do to establish a firm naptime/quiet time now will help you - and him - for some years to come!
Good luck!