As has been said, oil based and sanding. I really don't think the rest matters.
For automobiles, the main reason to use a good paint is durability (spray can jobs don't last long in the weather), but for an arcade machine that will remain inside, any oil based paint will last a very long time.
Five years ago I painted a PC case with krylon primer, purple paint (5 coats), and clear (5 coats). I sanded between each coat up to about 1000 grit on the final coats. It came out to a mirror finish and still looks that way (even though it's collecting dust in my garage now).
I've been using rolled on Rustolum on my cab and so far it seems to have the exact same results, though I'm not really going for a mirror finish on this one. I'll probably just use about 3 coats of paint with no clear.
The main key is just multiple thin coats of paint and tons of sanding.