I was thinking about buying an external hard drive and was wondering if it was possible to install a full version of Linux on it so I could simply just carry my OS around with me.
Any help would be appreciated.
It's probably possible. Try something like the following (NB. NOT TESTED):
Plug in external hard disk.
Run Linux install. Partition and install to external HD.
Install grub or lilo onto external HD.
Create a boot disk that will pass off the boot process onto this external HD.
Bear in mind that it probably won't work very well after it's done, because installed Linux distros probably won't like the changing hardware. Good alternatives include a live CD which will save settings onto a flash disk (IIRC Mandriva Move does this) or simply a laptop.
If you run a live distro it'll be slower as the computer needs to read from the disk. I recommend installing linux if you want to use for a long period of time.