Quit is grayed out in Mail
One of two things has happened to make Mail non-responsive during the normal quit process.


The entire program has hung (and you will see it as non-responsive in Activity Monitor or the Force Quit dialog window).
Mail's background threads for interacting with mail servers have hung and not timed out (and you can open the Activity Window from Mail's Window Menu) to see what specific account and perhaps even what it's trying to do (such as delete messages, send mail, update or synchronize a folder, etc…)


Once the quit menu is dim - you can either force quit the app, wait for it ti finish quitting or log out of that user (perhaps by rebooting) but you can be sure the Mail program is trying to quit at this point so you can continue to use it or decide to help it down as you see fit.