Don't use github/gitlab for small private repositories! use codeberg.org if you are not a developer. If you are in need for a big more jazz and looking at self-hosting go for forgejo.
Pretty sure codeberg.org uses forgejo under the hood.
Another very solid option for self hosting is just adding a git user to a server with git installed, initiate bare repositories there, then talk to them with git@example.com:repo-name