New Alternative
Because the given accepted answer does not work anymore, I thought I would explain how I was able to do it with the new changes in the github API.
The new Download Api Link
First, I found information about downloading the archive here:
https://developer.github.com/v3/repos/contents/#get-archive-link
Public Repo
If it’s a public repo then it is very easy… you can do:
curl -L https://api.github.com/repos/pengwynn/octokit/tarball > octokit.tar.gz
Private Repo
If it’s it is a private repo, you need to create an oAuth token by going to your settings and then selecting “Developer settings” / “Personal access tokens”.
I created a personal token.
Then using the instructions on the following page I found out how to get private repo that you have permission to:
https://developer.github.com/v3/#authentication
Full Code
curl -H "Authorization: token ab499f3b..." \
-L https://api.github.com/repos/godzilla/my_priv_repo/tarball > wut.tar.gz
Be sure to replace ab499f3b... with your actual token.