I'm having issues as well on a Mac Mini (Mid 2011) running Lion 10.7.5...
It allowed me to selected Mavericks from the App Store and it appeared it start downloading it. After a few hours of no progress, I rebooted in order to try again. Now when I select it from the App Store it allows me to "Install App", however, it just sits there and does nothing. On the Purchases page, it is listed with just a gray icon on the right - no wording. Doesn't appear to be doing anything. I can click it, but it doesn't change anything.
I've tried rebooting and shutting down. Each time I start over at the "Free Upgrade" and "Install App", then nothing.
Any ideas?