It's been 7 years since this question was originally asked, and it is still relevant. From what I've found, it looks like git-annex has been entirely removed from both GitLab CE and EE in favor of git-lfs. To me, using git-lfs is not an option, because that will force you to push the LFS data to every mirror, therefore it will take up the space on every mirror. As an example, if I wanted to have a mirror in GitHub, and I'm using LFS, it would mean that I'd need to push the LFS data to GitHub too, and this restricts me to comply with the LFS limits. In addition to this, git-lfs forces the user to download all the data on pull, and in my case we're talking about lots of data, so downloading all of it on every pull is also not an option. So for my use case, git-lfs is not a proper replacement for git-annex.
There is a pull request for adding git-annex to Gitea but I wasn't able to make it work.
I have only been able to make two self-hosted solution work with git-annex so far: (1) gitolite, which is not particularly easy to setup, and I eventually encountered some git-annex repo corruption problems that I haven't been able to fix, and (2) G-Node Gin which I was able to setup using docker-compose, although I had to deviate slightly (and simplify) the instructions provided in the website. Finally, you can always have a git-bare repo over ssh with annex initialized on it.