If you need to go to the hospital, you can call an ambulance that will be able to quickly and easily reach you due to there not being any car traffic. (The utterly ludicrous cost of US healthcare is its own separate problem)
Going to the countryside? Take a bus to the outskirts, or even out into the country itself, and cycle to a particular spot. Going to or from a club? Take the metro, take the bus, maybe even (depending no the strictness of ‘no cars’) a taxi, which you can afford on special occasions with the literally tens of thousands of dollars you’ll have saved by not needing to buy, insure, repair and fuel a car.
You have to also understand that for a car-free society to even be on the table, a number of other social changes will have to have been made too. So just arguing that going without cars is impossible due to the limitations of the car-centric society that currently exists is just circular reasoning. There SHOULD be ways to do the things you’ve listed without using a car, and the reason there aren’t is BECAUSE of cars.
What if an ambulance gets stuck in traffic? What if a road’s blocked by an accident? Or because it’s being repaired or relaid because of all the car traffic over it?
What if all the ambulances are occupied tending to one of the ~2,500,000 people injured in car accidents each year?