There are two main causes I generally see associated with the error message in question. The first is simply Apache having been restarted, which shouldn't occur all that often. The second is Apache running out of available connections, which is generally caused by a misconfigured or overloaded server.
In short I am fairly certain the problem is related to your hosting environment. That being said you did note your hosting environment is everyone's favorite choice for this (among other) exact reason. You should talk with their support as they might be able to resolve the problem, but for your long term sanity I would suggest moving the site to a more MODX friendly host.
If you are looking for a cost effective hosting option, as Sal pointed out above we are offering Ambassadors some free hosting. If you want a low cost option to test the site (confirm it's hosting related), feel free to take advantage of that offer. If you do go that route and the problem persists let us know, as I am more than happy to help you troubleshoot it further. It's much easier in a known MODX friendly environment, especially when the support has full access to the server and understands MODX