For that kind of monthly budget, you could get light grid/cloud hosting. Your flexibility would be, by far, increased over whatever system you might build. As well, the bandwidth is likely to be far better. Of course, there's always the 'unlimited space' options that nearly every host is touting for cloud-based services. Anything grid-based could mean the ability to spawn new VPSs at a moments notice.
However, IF you want to just build for your home, any decent gaming system is going to be great for web/media hosting. You don't necessarily need to waste $500 or so on a video card so you can have more money to spend on HDD space or something else.