I really don't know how to start explaining what is going on.
On my site, http://cldar.com/, I've got over 1,000 posts published, and when I'm logged in, I can see them on the front end of my site. But if I'm logged out, I can only see one. If I'm logged out, but I have a link to a single post, the post shows up, but if I try to browse the month's archive, it returns a 404. Categories work just fine too.
When I go to the edit posts page, this is what I see:
Which looks plenty normal, but when I click on the published link at the top I see this:
Also, in the first image, there's a post that's called "Photo of the Day – Roman" that's set to pending, I was trying to edit the content of the post and it wouldn't keep any of the changes, and I couldn't get it to actually publish and stop being pending. I've successfully edited posts and post content of other posts after finding issue.
I'm currently running the site with no plugins and on a fresh copy of twentyeleven.