If I were you, I'd start looking for another job ASAP.
Beyond that, here's how I handle my unmanageable email load. I don't think I get as many as you do, but I definitely get more than I can keep up with. If I read them all, I wouldn't get anything else done.
I scan my inbox periodically for emails that "look" important, based on subject line, urgent indicator, the sender, and topics in which I am already engaged and on which I am expecting an important message. Everything else gets blown off for the time being. If I happen to get an hour or two without any meetings, calls, or urgent issues to attend to, I might go back through recent unread messages and pick off some more that look like they might be consequential. But I never go back more than a week. If there was something important in there, the sender probably already called or stopped by my office, or complained to my boss about me not being responsive ;).
Right now I have something like 2,900 unread messages in my inbox, and the world hasn't ended yet. I've gotten the highest possible performance rating for the last two years, so I must be doing something right.