I have been having a similar problem for a few weeks and frankly, it is driving me nuts. I am pretty sure I’ve narrowed the cause of the problem down, in part to, a plugin (statpress_reloaded) that is too intensive on the MySQL server (the server load is perhaps my problem).
Anyhow, I can still log into my CPanel and then I go to phpmyadmin and click on a few random things, perhaps even create and drop a table, and the site comes back.
Pain in the ass, I know, but for a while I was deleting and reinstalling WP to fix the problem and this is much easier. I’ve disabled the plug-in for now as well and so far so good.