I've just tried everything and I cannot seem to fix this for the life of me.
Basically, Safari has started reporting the following error in the JS console: "SecurityError: DOM Exception 18: An attempt was made to break through the security policy of the user agent." Most of Ajax interactions via https are not working, so I cannot log into Twitter, Facebook or even Apple's own Developer Center.
I've tried installing Webkit, and even removing all preferences files that I'm aware of.
Other local browsers (Firefox, Chrome, Opera…) work just fine. I'm using Safari 7.0.2 in Mavericks.
Best Answer
Fixed. After inspecting the exact JS error when accessing the twitter login page, the offending line was trying to access the browser local storage.
The "Privacy" settings in Safari had the "Block cookies and other website data" set to "Always". I've set this to "From third parties and avertisers" and now it's working again.