I've put a list together here, and it totals $290 including shipping, but no OS
Like everyone else has mentioned, you will be hard pressed to come up with a system that will run games smoothly for $300. I think this build would play Warcraft on low settings, maybe?
GPU: Asus 4350--36.99+free shipping+$15.00 mail in rebate
http://www.newegg.com/Product/Product.aspx?Item=N82E16814121310
Wintec 2x1 GB DDR2 800 RAM--$45.99 + free shipping
http://www.newegg.com/Product/Product.aspx?Item=N82E16820161030
Antec 300 Illusion Case + Antec Basic BP 430 = $94.99
http://www.newegg.com/Product/ComboDealDetails.aspx?ItemList=Combo.315682
Athlon xII 240 + ECS Mobo--$106.98 + $18.00 mail in rebate:
http://www.newegg.com/Product/ComboDealDetails.aspx?ItemList=Combo.316100
With this combo, you'll wind up getting $33.00 in rebates. Windows 7 OEM costs $104.99 right now at Newegg. With the total being $290 shipped, that puts you at roughly $60 short of an OS, even using my "fuzzy" math. Your best solution would probably be to look for a used PC as others have suggested, or if you are a student, I think you can still get Windows 7 for $30, I'm not sure.
This is not a great build regardless. The mobo is not a great one. The RAM is just barely enough to get by on nowadays, and the PSU is not great either, but it should run that setup. Good luck.
EDIT: you could probably try to go with a cheaper case/psu combo, but even on a low budget build, I'd try not scimp too much on the PSU. You should be okay with a good 400 watt PSU from a solid brand like Antec.
I forgot to add a DVD/CD drive too
here's a good cheap one ($21.99 + $1.99 shipping) that I am using right now :
http://www.newegg.com/Product/Product.aspx?Item=N82E16827151175
That would bust the $300 budget right there...
Or, if you really wanted to cut some corners, you could get a pretty nice case with a not-so-nice 420 watt PSU for $49.99 + free shipping + $10 rebate:
http://www.newegg.com/Product/Product.aspx?Item=N82E16811156137
I don't even know if that case would accommodate the Micro ATX mobo though.
That would put you at $263.99 including shipping, minus $43.00 in mail in rebates.
Technically, you could say those rebates put you at around $220, only ~$25 shy of being able to get Windows 7 OS.
EDIT (again) I forgot a hard drive as well >_< so yeah, that would put you right back up around $285. Sorry