The Beginning of the End for the Old Mac?

So my older system no longer believes in the Intarwebs.

This happened at one point last week - when I try to connect using anything (browser, FTP, whatever), it says there is no such server. But after I re-seated the cable that connects it to the FIOS hub and restarted, it was fine. But now I've tried that 3 or 4 times, and it still can't find its way onto the Internet.

So I can't reply to e-mails I've already downloaded (I can write new ones, and reply to anything that's still in my online box), and to post anything I've been working on, I'll have to move it to this newer system. It's something I was going to have to do anyway, but I've let myself run out of time to do it at my own pace.

The e-mail is what I'm most worried about. I once tried exporting all my e-mail archives from the old Netscape it's in to the Mac's built-in e-mail, and it didn't work - it built the e-mail mailbox structure, but it was empty ... no messages.