Fedora project has created a simple script to install Fedora 7 on a USB Drive with 1GB+ capacity. To use the script you will need two things:
1. A working Fedora 7 installation. You can download the Fedora 7 DVD iso image from here. You will have to burn the image to a DVD and install Fedora 7 from it.
2. Fedora 7 Live CD. You can get the CD image for one of the two available versions here.

The follow the instructions in this document to install Fedora 7 on USB drive. You will also need to change your BIOS so the USB drive is the first booting device. After that you can even remove the hard-disk from your machine and work just from the USB drive.

Note: Booting from USB drives may not be supported on older BIOS. First check for BIOS support before you proceed. You can find if your BIOS supports it by trying to change the first boot device to a USB drive in BIOS settings. See if you can upgrade the BIOS to solve the problem.