Git – How to Backup a GitHub Repository

If you only want to backup project files (without history), you can use git archive (that way, I do not have to "watch for hidden file")

The script git-archive-all.sh will actually archive all git repositories and submodules in the current path.

Useful for creating a single tarfile of a git super-project that contains other submodules.

As Charles Bailey rightly mentions in the comments, git bundle is more appropriate, to preserve the history. (git bundle was introduced in February 2007).

See Backing up a git repository with git bundle

git bundle was designed to allow transfer of git commits when there is no direct connection between repositories (i.e.: offline) but using patches is not an option because of the large number of commits and multiple branches.
A git bundle is just one file which can be very easily created and again imported as it can be treated like another remote. A quick example:

jojo@dualtron:~/devel$ git bundle create ~/devel.bdl master test 

and a bundle is saved under ~/devel.bdl containing my master and test branches.
If I am now at repository B I just use jojo@dualtron:~$ git ls-remote devel.bdl which shows me the branches stored in the bundle.
To use the bundle I simple treat it like a remote, using git fetch (for example)

jojo@dualtron:~/git/repoB$ git fetch ~/devel.bdl refs/heads/\*:refs/remotes/bundle/\*

The other official way would be using git bundle

That will create a file that supports git fetch and git pull to update your second repo.
Useful for incremental backup and restore.

But if you need to backup everything (because you do not have a second repo with some older content already in place), the backup is a bit more elaborate to do, as mentioned in my other answer, after Kent Fredric's comment:

$ git bundle create /tmp/foo master
$ git bundle create /tmp/foo-all --all
$ git bundle list-heads /tmp/foo
$ git bundle list-heads /tmp/foo-all

(It is an atomic operation, as opposed to making an archive from the .git folder, as commented by fantabolous)

Warning: I wouldn't recommend Pat Notz's solution, which is cloning the repo.
Backup many files are always more tricky than backing up or updating... just one.

If you look at the history of edits of the OP Yar answer, you would see that Yar used at first a clone --mirror, ... with the edit:

Using this with Dropbox is a total mess.
You will have sync errors, and you CANNOT ROLL A DIRECTORY BACK IN DROPBOX.
Use git bundle if you want to back up to your dropbox.

Yar's current solution uses a git bundle.

I rest my case.