1. Click on the little person in the upper right hand corner.
2. Scroll down to your blog and click it.
3. Click Edit Appearance on the right hand side.
4. Then scroll to the very bottom you should see Export and the name of your blog. It will begin the backup process.
5. You should then receive an email from Tumblr telling you that your blog is ready to download.
6. Go back to edit appearance, scroll to the bottom and click Download Backup.
7. If you have side blogs you will have to do this for each one.
8. You can not do this on mobile.