[jdc]

Try GitLab.

[arno1]

Or Gitea!

[zeckalpha]

Do either store issues as objects in the repo itself?

[cmrdsprklpny]

Not git, but I believe fossil[1] does this.

[1]: https://www.fossil-scm.org/

[jto1218]

Do you know of any systems that do this? Seems like a good idea.

[CrazyPyroLinux]

I really like https://github.com/gitbucket/gitbucket for this.