Is it possible to build your own streaming site? I’m trying to build something so I can stream directly to my website instead of having to use mixer/twitch etc

Well, as proven by twitch, it's not impossible.

Id be willing to bet there's an open source streaming project out there somewhere.

If you intend to stream for creative audience, linking to Picarto may be more relevant than reinventing the wheel. But maybe also search up for something like "HLS streaming server open source". Cannot really help other way you there 🤔

https://webrtc.org/

Don't say I never gave you anything 🚬

You need a server that can handle the transcoding and compression + god tier internet connection depending on the amout of users.