Just download the ISO for Ubuntu: Get Ubuntu | Download | Ubuntu
Burn that to a DVD, and boot off of it, and choose the LiveCD selection in the menu. Plug in your external HDD, and open up the file browser on the LiveCD environment, and then you can copy/paste files from your internal drive where all your files are stored, to the external drive.
Your external drive should work since the USB drivers will be loading off of the LiveCD. The only way they won't work, is if you have an actual hardware issue with your USB ports.