I don't believe it is possible, no. Even in places with frequent trains during the day, they typically stop at night. Say I need to take someone to a doctor and fill a prescription at 3AM. The usual pharmacy and urgent care are closed. Is there a city on earth that has trains to exactly where I need to go, every five minutes, at 3AM? Will the train get me there as fast as, or faster than, a car?
IMO transit is incredibly useful for situations where everyone is doing the same things at the same times. But it forces you onto a schedule and a destination, with little flexibility.
Privacy, for one thing -- I don't think I'd ever want an unknown third party in the vehicle with me when I'm taking someone or myself to or from an unscheduled medical visit.
The reasons I own and drive a car would be impossible to meet by anything except self-driving personal transit pods that can link up with other pods to form dynamic trains. That's the sci-fi future I'd like to live in, where I have a private pod, but if I want I can connect to a train for a long distance ride, use the dining car, meet other people along the way, etc. Kind of like how ferries work now, except you could stay in your car if you want, and faster.
I suspect it would push the cost of driving out of the reach of your average person though.
You often see this (in the UK) where people living in London assume that everyone else has 24 hour bus services.
See what he means. There is no reason transport needs to force a schedule life on you. Good transit is frequent enough that when you decide to go you just walk out the front door to the station and wait for the ride to show up - it will be soon enough.
But you have never seen that so it hasn't occurred to you that it is possible.