It has GPU requirements there that are similar to the Windows side, still won't allow the use of a VM, and will still need a lot of CPU and bandwidth. The host OS is less relevant, as the Linux version of OBS can't be run headless; even if it could, you'd still need an X session to show the Osu! window and capture it.
I'd say that $80/mo would be on the cheaper side for a dedicated non-virtual server with the necessary GPU, plus a beefy CPU, plus the (significant) bandwidth fees to run a stream 24/7. Unless you're looking at an unmetered rate quarter rack and providing your own server hardware, which will end up well into three, if not four digits per month, depending on the rate at your local DC/NOC.
Now I'm tempted to see if I can get OBS-MP running on a PowerEdge 2950 under CentOS or Debian. I doubt the built-in video card will be able to swing it, though.