You could in theory build an arcade machine for $50. Free TV, free PC (or free/cheap console; I got an XBOX for free once), wood isn't that expensive. But it would be extremely lacking, as Beta said. Yes, not every arcade machine has a marquee or sideart. But the vast majority do, and nothing says "homemade ghetto job" like missing those things. And what about controls? For that budget you'll only be able to use the crappiest used flux imaginable. The only possible way you'll be able to get in under $50 and have real arcade controls is by finding a half-gutted machine on the side of the road, or get lucky on eBay. For Candytop the controls ALONE cost $100, and if I went cheap it worked out to a bit under $50 IIRC.
So yes, it's possible, but if you need to buy tools, break something, can't find something cheap or for free, the budget goes out the window. That, and you'll end up with a substandard machine that doesn't impress.
Most arcade machine builds are in the same price range as a portable, but the really impressive ones can cost well in excess of a thousand dollars.