I have the same issue with one of my Apple IDs.
I use 2, an Alerican one and a French one.
Everything works fine wirth the newer French one.
But I have had issues with the American one for several weeks.
One is that I get an error message signinf into iTunes that the iTunes store is unavilable. Deleting cookies and certain files is a temporrary fix.
And for the last few days (I was already using Mavericks) I have the same issur as you signing to the Mac App Store with my American Apple ID. (load for ages).
I have spent several hours with iTunes and Applecare teams over the last month and a half. They provided a temporary fix for iTunes but nomore.
The issues are the same on 3 different Macs, as well as Windows 7 inside VMWare (except of course that there is no Mac App Store on Windows)
Anyway, I have a temporary fix for accessing the Mac App Store :
- Create a new user in System Preferences/Users and Groups. A standard user is sufficient.
- Login to your Mac as that new user.
- Sign on to the Mac App Store.
- Buy / update your apps.
- Log out and delete the "new" user in System Preferences/Users and Groups
I could see that if I sign on iTunes with my French Apple ID (as the new Mac user), I then have the same problems when trying to sign on with the American ID.
Hope this helps,