If you only have $300 to build a pc your going to struggle to play most games with a system based in that price range.
Even if you could, you would be looking at playing something like Fallout 4 with very low screen resolution and with just about all of the in game graphical effects of the game set to low.
For about $300 your best bet is a PS4, in fact the game graphics would be better than what a $300 pc based system could produce, and would play a lot smoother of course.