JD can certainly correct me if he wants, but I think he was being sarcastic.
Any 24 year old car in DR has likely been driven to death, and will either be in need of serious repairs (or just need to be outright rebuilt) for the sort of trip you have in mind.
I'd be very, very careful here. The last thing you want is unreliable transportation requiring parts that, because of the vintage of the car, may have to be sourced from bigger cities.
Seriously, consider public transportation. The buses will get you to the bigger towns, the smaller vans go to or through every populated area on the island. You can then hire a car, or get around with motos. It's safer and more reliable, far cheaper, will eliminate the chances of you getting lost and broken down on some lonely road, and will give you a truly authentic experience.