Trying to do this using Outlook or Outlook Express would be an enormous PITA. Maybe not even possible depending on security restrictions some of the systems may have in place.
However, you're in luck with Portable Thunderbird. It's a version of the Mozilla Thunderbird email client designed to run off of a USB flash drive. Meaning all your email messages will be stored on that flash drive along with the program to access them. The fact that Thunderbird has far better provisions for dealing with spam, is less susceptible, to email based worms and viruses, and all the other handy features are just a nice added bonus.
You can also get Portable Firefox, so you can have a consistent set of bookmarks, extensions, cookies, saved logins, etc for all your web browsing needs.
The other viable option would be to get a web based email account, such as Gmail, Yahoo, Hotmail, etc, and divert all your email to it. Then all you need is a web browser to access your email.
How can I copy my Outlook emails to a flash drive? I bounce from computer to computer, most of which are not connected to the net or networked. I want to be able to temporarily access my e-mail from any of these computers without the files being installed on every computer I use. Ideally, I would like to have Outlook automatically save a copy of my e-mails to a flash drive. Can this be done without buying additional software? If not, what software would you recommend?