As outlined above, Github pages do not support that functionality. I had the same issue when our team decided to host project documentation (static HTML) internally and privately.
I ended up creating a service https://www.privatehub.cloud It is basically a simple proxy server with Github OAuth authentication, so it merely returns your GitHub repository content with a proper MIME type. By design, only who have access to foo will be able to see foo content at https://bar-foo.privatehub.cloud. From functional point of view, you can think about it as a simplified GitHub pages with built-in authentication.
Unfortunately, Github OAuth does not allow to request read-only access to private repos, so the server needs the full access (obviously, it does not write anything to your repo). As GitHub API allows to retrieve files under 1 Mb only, the service cannot return larger files. Yet, I found the service is quite suitable for small projects for internal documentation or staging version of a website.