Can I download Linux onto a flash drive, say a 1 gig size.

Yes. Here is a great site just for that:
http://www.pendrivelinux.com/

PCLinuxOS has been working on pen drives for a while. I think they even sell a pen drive with the OS already installed, but doing it yourself isn't hard, and 1 gig is all you need. Bigger gives you more options, but 1 gig works.

Here is a tutorial for doing it:

Pendrive installation and use