Can't connect to *one* specific site anymore.

Are you running a load balancer? Or proxy by any chance? The ip might of been blacklisted. I would check with your isp to see if normal traffic has been automatically blocked due to the proxy looking malicious - sometimes and usually when they update the server sees and treats the traffic coming from the proxy as individual traffic and automatically blocks it because its trying to call too many pages at once instead of what it is and then sees normal traffic as the server. It basically swaps the two and blocks the ip - your standard tier help person will not know how to check for that and unlist you. I just had the same problem with wpengin and was getting a 502 error. Hope thats it - its simple enough to remedy at least until the next update.