After going through various apple forum posts and deleting plist files and caches, nothing seemed to work. Then I came across a post that suggested that the problem could be with the Ethernet controller.
This is the error I was getting. All it said was "An unknown error has occurred", the Console didn't have any other messages, overall, not very informative.
Below are the steps on how I managed to resolve it.
Turned out the problem was the Ethernet controller after all! In my Network preferences, I've had my old Ethernet controller set up (since I migrated my Mac over from a MacBook Pro), this was no longer functional of course.
After deleting the Ethernet controller (I also removed the FireWire network and the modem that was configured), I've restarted. The next thing to do was to add the actual Ethernet controller that's in the computer now, unfortunately it did not show up. Oh well, since this was a Hackintosh, I went back to Multibeast and installed the Lnx2Mac Realtek driver and restarted again, this let me add a new Ethernet controller to my network setup.
I set the IP address to manual just in case.
Getting back into App Store and logging in, everything works now!
It's strange for App Store to require that Ethernet is configured even if it is not being used.