Your Mortar projects are stored in GitHub and are accessible to all users in your Mortar account. To find the project in GitHub go to the Projects Page in Mortar, hover over your project, and click the "View Code" eye icon on the right.
The best way to preserve your project is to create a copy of your repository in your own GitHub account.
To move your project to your own GitHub account follow these instructions for Importing a Git repository using the command line.
If you don't want to move your project to a different GitHub account, you can clone your repo locally to preserve it. To clone your repo locally, use the SSH or HTTPS clone url found on the right hand side of your GitHub project page and do:
git clone YOUR_CLONE_URL
All of your repository history, tags, and branches will be preserved in your cloned version of the repository.
You can use the "Download Zip" button on the main GitHub page for your project to download a compressed copy of your current code. However, this copy will not contain any of your commit history or code that is in other branches.