Okay, so first of all, let me give you a brief overview of how the Internet works. You have your computer and it is given an Internet Protocol (IP) address by your Internet Service Provider (ISP, the company you pay for Internet access). When you type in a website (like www.yahoo.com), that gets converted into another IP address (209.191.93.53). From there, it's like a phone call- your computer is talking to one of yahoo's computer's, asking it for information. All of the information is stored on Yahoo's computers and Yahoo can just keep adding more hard drives.

So no, the Internet will never run out of room because it consists of millions upon millions of hard drives and all you have to do is plug in another hard drive to get more space. There currently is a project called the Wayback Machine that attempts to archive the entire Internet from 1996 onward. It is currently 20 petabytes (20,000,000 GB) and it grows by 20 TB (20,000 GB) every month. So, since the archive is also on the Internet and I don't believe it has the whole Internet (password protected stuff probably doesn't get added), it's got to be much bigger than that.