well you can buy a good gaming video card for about that much, but a whole system is not going to be good for gaming at that price. unless you dont mind playing older/less demanding gameshttp://pcpartpicker.com/parts/partlist/
that is about the best your going to do for $300. that is bare bones no OS, no keyboard or mouse, basic power supply. It will run most of the latest games in low-medium detail settings, demanding games like bf4 or crysis 3 it will struggle even on the lowest settings.
you could put a better i3 cpu and 2g 7790 gpu bring it up nearer to $400 http://pcpartpicker.com/p/2rUKm
and you would have a reasonable gaming computer. if you raised it to $420 you could get a 2gb 650ti boost which would be a fair bit better, good enough for 1080p gaming at medium-high detail.