The number one google result is Owncast.

Looks like it doesn’t save the video by default but there are workarounds to record it.

To add to this a bit, if the site is just a single page that will be fine. If your site has a bunch of pages that all need to share the same theme and headers and footers and such it may be worth learning a simple templating system that runs before the site is sent for hosting.

That way if you need to update something in the design or theme you can do it once, rather than on dozens of pages.

Personally I use eleventy for my little site.

https://www.11ty.dev/