Yes. You can use Ubuntu as a LiveCD, which means you do not have to install it right away to use it. You can transfer files using the LiveCD.
Here's how:
Burn the disc as the installation says, then use it to boot the computer. The computer should recognize the disc and boot from it.
Now, on boot you will see a language selection screen. Choose the language.
You will see option on the menu. Select the one that says "Try Ubuntu without any change to your computer."
It will then begin loading Ubuntu from the disc.
You will then see a "human" background. Select at the top "Places" and then down to Computer.
It will display all of the drives. You can freely access that data and transfer it to an external drive or flash drive. It will probably be labeled "00 GB Media" - that is the hard drive. If you see another drive in there, like a flash drive then it might say "00 GB Media" as well. Try to tell the difference. (00 is just a generic idea for the number of GB in the drive)
Plug the flash drive in, and copy and paste the files you want from the hard drive on to the flash drive/external drive.
Ask questions if needed.