Getting Error 324 (net::ERR_EMPTY_RESPONSE) when saving changes to content/views/modules
So, after a few days of trial and error, I managed to fix this problem. (Although I still am not really sure what caused it...)

The fix that I found was to disable modules until the problem disappeared, then to re-enable those modules. That was it. The problem has not resurfaced for the past few weeks.

Now, some details: I first attempted to fix the problem on my "sandbox" server, disabling modules one at a time with the hope that the problem would go away. It did, and I noted which module(s) were the last to be disabled. However, taking that information back to my main site did not help. Instead, I disabled modules in bulk (everything that could be disabled on a first pass; i.e. all modules not required by another module). After a few passes, the problem did indeed disappear. I'm not sure what caused it or why. If my searches while trying to fix it were any indication, nobody else knows either. Just a quirk of Drupal 6 I suppose.