Well, a personal archive of the very interesting things of the internet circumvents the possibility of removal... I can't keep everything, my "big brain" hard drive only has 120 GB, but if you have at least some interesting files stored, everyone can throw them onto a big heap and that's the old internet! Well, part of that at least.
Maybe, some day, the computer freaks have to set up their own wireless router, participating in the local city mesh network. But that will be called "illegal" then, so routers will be hunt down by drones searching for "unauthorised" electromagnetic waves...
What's left will lack the transfer speed, if the "offical infrastructure" of wires can't be used. The skill of optimising file sizes will be important. The only online game will be Nethack. The underground video portals will transmit their movies as 5-second GIF, no previews, only direct download. IRC should be doable...