|"Supposedly" running from a USB drive (assuming you can boot from it) can be done:|
But I'd be a little wary of both where he got the Win98boot.zip file and multiple problems mentioned in the comments. Use at your own risk.
Running Win98 from CD would be nearly impossible, since the CD files would be "locked" and nothing could be written to it. There is, however, a Win98 Live project:
Personally, I agree with the comments above. A live Linux distro (I prefer NimbleX), would be a simple download and burning of the ISO file to CD. If the machine has enough RAM, you could even run the entire OS from RAM (though the "386MB" you mention in your specs would significantly limit your choices).
But it could simply be less trouble (and maybe even cheaper in the long run) to get a new hard drive.
"Channeling the spirit of jboy..."