Can I machine polish my car myself?
You certainly can, and many people do. In fact, it is a popular pastime with many enthusiasts out there who spend their weekends polishing their cars, their friends cars and their family's cars, because they enjoy doing it. Some even make some money out of it.
In fact, it is fair to say, that because they do it for fun and not for profit, they can put way more hours into it than a professional. Whether they get better results is debatable and depends on the skill level of whose doing it, but detailing enthusiasts can do amazing work, especially those involved in custom cars.
So if you want to do it, because you think you will enjoy it, then you certainly can and there are many online forums and Facebook groups dedicated to the hobby which will help you get started. (They may also hinder you, because like any online community, there is elitism, factionalism, fashions and trends, bad advice, egos, vested interests etc.)
There are also companies who will offer courses teaching you how to do it, or a cheaper alternative, there are YouTube videos which will give tutorials.
It might all get expensive, though. If you are paying for courses, random orbital polishers, polishing pads, a range of compounds and polishes and then waxes and sealants, cloths and towels, it will soon add up. So if you are thinking for polishing your car yourself to save money, that's unlikely to pan out.