Maybe an 8 gig memory stick would work but it might also need a driver to read it.
Can you open up the computer and remove the hard drive and keep it?
Can you email some of those files to yourself? Installed programs can not get transferred off anyways. So just Word, Excel documents. pictures, and email the more important ones to yourself. Each email will take time to send out so you might need to get like a gmail email account and stay up all night emailing them out.
USB technology came out the same time as Windows 98. Even a USB memory stick will have trouble. And it sounds like you have no time to buy a memory stick from a 24 hour store. Even then you have to look for a Win98 driver if it does not recognize it. The external hard drive might not be working because it can not draw enough power so maybe a small memory stick would work.
Update: I read your other question on this. So the old Win98 machine has no internet so you can't even email off of it.
And if you have no friend to help you remove the harddrive then look at some youtube videos on it.
http://www.google.com/search?hl=en&source=hp&q=youtube+removing+harddrive
Then your best bet is to see if a memory stick would work on it. It does not need much power as a external harddrive.
A old Win98 machine might have a cd drive that can burn CDs. I doubt if it can do DVDs. That technology is newer. Maybe buying an external cd writer would work but not if it goes through a USB on your machine.
Do you have a 24 hour store that might sell memory sticks?
Again, you might have to surf for a driver for it.
http://www.google.com/search?hl=en&source=hp&q=win98+memory+stick+driver
And the manufacturers website for the memory stick is the best bet for a win98 driver. You never said the brand of the external drive so I can not even look for a win98 driver for you.
And for USBs, Windows 98 SE (Second Edition) reads USB devices a lot easier than plain Windows 98. Do a Start - Run - explorer.exe and then do a Help About Windows to see which Windows 98 you have.