Obsidian doesn’t really have self-hosting as far as I know, even though it is an electron app so it would be trivial for them to do, but it would cut into their revenue stream of obsidian sync. One of the downsides to closed source. The only things related are static website “wiki” generation from the markdown file collection.

Quartz github.com/jackyzha0/quartz

Or people use it for blogging like

Hugo and Jekyll

Syncthing (or nextcloud/Opencloud) can be used to sync obsidian vaults across devices though.

I would suggest opencloud over nextcloud as it is quite a bit faster and not just a mass of extensions and tends to break a lot less than nextcloud. I stopped using nextcloud because of the amount of times it completely broke on an upgrade while not even having breaking changes (I’m glad I didn’t have end to end encryption)