No Dashboard RSS feeds and internal error 500

One of my WordPress sites that has been running flawlessly for 3 years now suddenly started exhibiting strange behaviour. At first I noticed that Jetpack stats aren’t available anymore – when I try to access them it attempts to load for quite a long time, but in the end comes out with a message to try again, because the stats couldn’t be displayed. After that I noticed that none of the RSS feeds that normally display in the Dashboard (WordPress Development Blog, Other WordPress News, Incoming links, Plugins, etc.) do not display any content, just messages “WP HTTP Error: connect() timed out!”. The updates page doesn’t work, returning a Server Internal Error 500 instead. So no available updates are either being displayed, either for plugins, themes, or WordPress core.

The strange thing is that this happened suddenly – a few days ago it was ok, I was away on a trip without internet access, and when I came back this started – absolutely NOTHING changed in the WordPress configuration in the meantime, nobody was installing or altering anything.

I contacted my hosting provider, but they claim that it is not a problem with their server, but a problem with my WordPress installation. The funny thing is that I have two more WordPress sites hosted with the same company (but on a different server and different hosting package) which still work flawlessly.

The affected site (www.stazeibogaze.info) is a multiuser site, and the number of articles has exceeded 500, so I wonder could this be caused, perhaps, by the database growing beyond a certain size? Or maybe the problem could be causing a timeout that is due to my slow ping (since I live on the other side of the world from the server location)? Do you have any idea what I should alter in my .ini, .htaccess or any other configuration files to correct this issue? I’ve already tried turning off Jetpack, but it is obviously not the cause.