I'm Not Sure it Will Work, but...
I'm not a computer genius and apparently what you want is very difficult to do, judging from replies to similar questions online.
My best guess from what I found is to create a bootable image of Linux on a bootable CD, then boot from that in your harddrive-free laptop.
Here's a video of a guy doing that with Ubuntu Linux. The comments below the page tell you how to create the bootable image and CD.
On the bootable image, you should have Google Chrome installed, with the Whitelist extention installed and set up the way you want it. Whitelist blocks every website except for the one(s) you "whitelist". I don't know if it's a free extension or not, but it's the first solution I came across, so there you go.
Whitelist link from Chrome store:
Like I mentioned before, I'm not a computer genius, and I can't guarantee that a savvy user would be 100% restricted from cracking their way through this setup and into the "I Can Haz Cheezburger" network, but it's the best solution I could find.