I did this recently to start creating a public (but unlisted) developer log, à la Carmack .plan.
Obsidian has a community GitHub backup plugin, but you can also just manually push to a repo whenever you want to update the site. Then you can use that repo as a proto-CMS, and pull it in with, say, NextJS to create a static site. Anything with built in CI (Vercel, Netlify, GH Actions) should be able to detect the change to the CMS repo and rebuild your static site. Both steps are quite easy (took me under 10 minutes total).
That said, if you don’t need special control over your content/presentation, then you could alternatively just pay Obsidian $8 a month for their Publish feature.
I feel like it should be easy to make Obsidian the content editor for a static site generator like Hugo, but I haven't found a tool for that yet. (Maybe it's so obvious that one isn't needed, but I'm not a dev, just a normie)