I'd imagine that a few of us with some natural talent, and a couple years of racing experience, could drive a NASCAR car around the track and look the part, after having 40 hours of seat time in the car/chassis to be used.
"Racing" one, and being competitive, would require 100's of hours of additional seat time
I raced for few years in the NE region at Lime Rock, Pocono, and Watkins Glen
I progressed from my Novice SCCA license, to my "Full Competitive" SCCA license over the course of three summers. I competed in the " American Sedan" class (AS), production modified Ford Mustangs 302 CID. I'd need quite a bit of practice to get back there today, and these cars were tame compared to a Sprint Cup car.
Just to get a Novice permit requires an extensive physical exam, and a review of your medical history. Then you need to pass the road test in the proper vehicle, at a SCCA Drivers School. Rentals are sometimes available for $750-$2000 for the test.