I've slept in the back of a rental car a couple of times (although rented for 2-3 days).
First time in San Juan, PR, when I had an early flight and was out kayaking on the east of the island and wasn't going to be into the city until after midnight so paying $200 for a hotel for 4 hours didn't seem worth it. Tried to sleep in the car but felt too exposed and not that comfy so just went to the airport, dropped the car off, and sat in the terminal for 3 hours or so.
Second time, I arrived in Honolulu late and by the time I picked up my car, it was close to 12:30am so I drove to the beach and parked up. Then drove to Pearl Harbour around 5:30am.
Have pulled over to the side of the road a few times in my own car back home and also stopped off in a motorway services car park before for a break / doze.
Suppose it helps if you know in advance where you can park.
Also, not sure if it's different for Americans as us Brits need to pay for various insurances and can't just use our normal car / auto insurance and that could push the price up.