No, absolutely not. Even if you could buy the 30-50,000 parts individually, the markup alone would absolutely kill the feasibility, much less the ability to weld the frame components together, assemble the literal miles of wiring, or program the computers.
Getting the chassis would be hard, usually that's not a "part". But you could by a second hand/wrecked car and refurbish it with parts. It will be expensive and not easy, but possible.
I love these kinds of questions! Car culture is so extremely broad, you need to narrow it down a lot to get a fitting answer. What car are you talking about?
In that case: it might be possible, but it would cost an insane amount of money. It would not surprise me one bit if the cost would exceed $100k. And it will matter little whether this is a Camry or a Honda Jazz or Protege or whatever. The problem is the insane markup on individual parts, not to mention markup by the dealer. And then you have to also pay for a space to put everything together. It the end you’ll end up with a janky version of a car which probably has a metric ton of issues, for the most outrageous amount of money.
If you really want a “new” old car, your best bet is to scoop up a low mileage version. They will be overpriced, but will still be a lot cheaper than the initial idea. I’ve seen listings of ‘80s cars with < 100 miles on them for sale. Always extremely expensive, but there are people interested in that kind of thing. Do note that a 10+ year old car that has not been used, is going to require work to get going. Rubber deteriorates, things seize when not used.
If your initial idea came from the wish to save money, just get a decent second hand one and be happy with ever year you’ll get out of it. Learn to do maintenance and you’ll keep running cost down and the car will likely even last longer.
There are some cars that are sold this way, apparently. They built one at the start of an "expedition" episode on the Amazon Prime version of Top Gear. (Is the show called The Grand Tour? Something like that.)