It’s just a bunch of markdown files. You can sync with anything. I just use a git plugin.

As others have pointed out you don't "host" Obsidian - it's just a local collection of markdown files. But if you're asking about a self-hosted alternative to "Obsidian Publish" for creating a knowledgebase that others can answer, I'm Quartz[0], a static-site generator designed to turn Obsidian markdown files into a website. I'm building and it and hosting it on Gitlab Pages at work[1].

[0]: https://quartz.jzhao.xyz/

[1]: https://quartz.jzhao.xyz/hosting#gitlab-pages

Thanks, I just need it locally on my own machine, do t need to share. Just need to back it up. I can't install software, but I can run containers, so I thought I'd just run osidians docker image and host on my mavhine

I recently went through this effort and I'd say it's worth it. In particular, I've used the "Self-hosted LiveSync" plug-in + docker.io/oleduc/docker-obsidian-livesync-couchdb, and I setup all my infrastructure within a VPN (with Tailscale, this was pain-free).

I knew about using just git, but having Windows, Linux, macOS, iOS and Android, it was just way easier to use this solution instead of fighting with git-like apps for each OS.

With Syncthing you don't need hosting. I use it for everything I can.