No.. not everything is classified and there’s different email servers and requirements for classified information.
On a classified server, emails are marked as such based on their classification but that doesn’t mean you can’t send classified information over a non-secured server which is what Clinton appears to have done and her BS story about never sending or receiving anything marked classified could be easily true but just because it’s not marked as such doesn’t mean the information isn’t classified. It’s the content of the email that makes it classified.
Much of what ANY Secretary of State deals with, if not classified, may be considered extremely sensitive, and its disclosure may result in serious disruption of diplomatic relations.
This is why much of the information is transmitted by courier and NEVER entrusted to electronic transmission, no matter how urgent the situation. While the courier COULD be intercepted and the documents stolen, it is easier to guard against such a detainment than to plug all the potential leaks in the Internet or other forms of electronic data transmission.
There is no marking named "classified", so her story is true but misleading.