If you’ve got a decent broadband connection, you can get one of many Web-based file storage services to park your files on - just upload them from the old PC and suck them down onto the new one. Of course, if you’re able to use a broadband connection on both PCs, you’re probably able to network them, in which case, you ought to be able to bypass the Web storage entirely.
Emailing stuff to yourself is another tried and true option - round up small files into ZIP files to contain and compress them. Again, this will be limited by your internet connection. Tolerable with broadband, but it’ll be miserable with dialup.
What is the old PC? You might be able to temporarily intall an internal CD-RW drive - basic drives are disposably cheap now and can be had for under $20.
One last thought - what are the files? If your old PC is so old that you’re having problems moving the files, you may well have problems with the files themselves. eg: obsolete versions or even obsolete applications that won’t run on WinXP.