Mac keeps asking for email passwords

I'm having an issue with my Mac continually asking for my email passwords. This occurs with my gmail email account, my @me email, as well as my @iCloud email. The problem began with all accounts at the same time even though I never changed my passwords. I'm not having any issues on my iPhone and I can log into my iCloud account online using the same password I'm trying to enter in my email account on my Mac.

The above screenshot shows I deleted my gmail account and re-added it but this is where that process got stuck. I unchecked contacts, calendar and notes then clicked done and its been stuck here for a good 10 minutes.

I then deleted my @me account and tried to re-add it but that came to a dead end. I entered my email address then my password for that email address then clicked next but the process came to a dead end at that point.

I can't delete my iCloud account because it keeps asking for my password and after I enter that I get an error message stating it can't verify my email and password.

I believe this originated with a keychain problem but I'm not sure how to fix it. Any help would be greatly appreciated.