A computer that’s on the public internet to run it on. While crunching through video and serving it to viewers can be intensive from the computing side, you can get away with pretty meager resources on a simple setup. If you don’t already have a server to run it on you can get a Linode instance for $5/mo that runs it fine. If you worry that you’ll be maxing out the bandwidth or transfer limits allotted to you, then utilize a service like Media Network to serve the files instead.
Here are some hosting providers people have been successful with.
Paste the following into your shell and the installer will download the most recent version of Owncast for your platform. It will also download a copy of ffmpeg if you don’t currently have one installed.