I asked this question on Facebook and Twitter, and got a variety of answers.
I think there are two "correct" answers.
Start a new Internet.
Cry.
Most people say they would go do normal things that don't involve the Internet, but I think the world would take a long time to adjust. Many of us remember a world without the Internet, but we don't remember a world without the Internet that used to have the Internet.
A question that reveals the problem is to wonder what would happen if you woke up one morning and found there was no electricity. Not much would happen in the world as it's currently configured without electricity, even though there was a time when it worked fine without it.
I think the Internet is sufficiently integrated into our civilization at this point that if it were to be removed, it would be such an enormous shock to our economy that.. well, that's why #2 is also a correct answer.
I'd like to think we could quickly boot up a new net. That's more likely than rebooting electricity because a lot of people who built the initial Internet are still alive and would know how to layer things so it could come back pretty quickly. Assuming the electric grid isn't dependent on the Internet. Which is itself a good question.