Your budget and your goal are so far apart you have 3 choices
1 - VASTLY Lower your goal
2 - VASTLY increase your budget
3 - Run gameservers so utterly awful nobody uses them (and 24/7 uptime is a pipedream)
If your doing this to make money you're a cowboy and one that will fail I suggest you find another scam. If its not to make money then I hate to say your dreaming
You might be able to make a pentium based box on your budget that would suffice for a basic COD or minecraft server uptime on it would be ungaruanteed due to cheap parts.