Is it possible to download only a sub directory from a git repository?
Or download a specific commit's state?
I read about tags, but the repository which I would like to download does not have any tags...
Since it looks like you have two separate questions, I think I can address both.
1.) Download only a subdirectory from a git repository
git fetch
git checkout HEAD directory/subdirectory
2.) Download a specific commits state
You can do a hard reset using the commit id. (a hard reset will discard any local changes and bring your repo back to the state of that commit)
git reset --hard {commit_id}