Okay, I admit I've never done this before. And since my parents inexplicably refuse to secure our network, I'm not going to try it now. But I can at least explain the first part...

You'll need to start by enabling file sharing on your network. I'm not sure whether both computers need to enable this or just the sending computer, but if you're just doing a file transfer you may as well do them both for now. (Hopefully you're using Windows, because I really have no idea how to do this with a Mac.)

Go into your Control Panel and choose Network and Internet, then go to Network and Sharing Center (or whatever similar section exists on your OS). You should see that "Public Folder Sharing" is unchecked; enable this and any other applicable options like "File Sharing," "Media Sharing," etc.

If I'm right about how this works, you should be able to access the old computer's files and folders over the network now. Then it's just a matter of copying and pasting them permanently onto your computer.

...Keep in mind that I'm looking at this with Vista, and that I've never actually tried the full process before. So you should probably take this with a grain of salt. But if you want to avoid buying external storage devices, I believe this is your best bet. May as well give it a shot, right?