The problem with a flying car that hovers without wings or rotors is gravity. To stay aloft, it must continuously generate enough upward thrust to counteract its entire weight. Unlike a car sitting on a road, it can't rely on the ground to support it.
Rockets can do this, but they must continuously burn propellant to generate thrust. A 2,000-pound vehicle would need at least 2,000 pounds of thrust just to hover, and it must carry both its fuel and oxidizer. Even under idealized assumptions, a vehicle carrying 10% of its initial weight in rocket propellant would hover for only about 26 seconds.
Jet engines and ducted fans offer other possibilities, but they still have to move air downward to generate lift, consuming energy in the process.
So the question isn't simply whether we can make a car fly. We can. The question is how to generate enough lift, efficiently and safely, while carrying passengers, propulsion equipment, and enough energy for a useful journey. Gravity isn't a minor engineering inconvenience. It's the bill that has to be paid every second the vehicle stays airborne.
This is undoubtedly why the flying Tesla is a joke.
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