Depends on what your budget is. The file hosting part of it would benefit from having raid disks down the line when you're expereicing slow downs. Before that Dual Core or Dual Processors systems are ideal for this.
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At a budget of $120, your choices are so limited, the question isn't so much about the best server to run that particular script, but the best server you can get for $120, period.
Basically, you're looking in the single core Pentium or Athlon range, maybe around a P4 2.8 or 3.0, an Athlon 3500 or 3800, 120-250 gig single SATA drive, a gig or two of RAM, and maybe 1000-1500 gigabytes of bandwidth. And this is at the outside, self-managed, and without a control panel.
It's going to be somewhat slow on the recompression of video, but even YouTube does that in the background, stating that it will be a few minutes for a video to be processed and go live after submission. So if your submission volume isn't too high, you're not in bad shape with those.
The real issue with video, though, is that you have to be very aware of bandwidth. If you have one 5-minute video at 160kbps being viewed by 4,000 people a day, you're either going to end up going over your included bandwidth or you'll be getting close to it, and the overage charges could add up.