Verify that someone didn’t walk in through an easy to guess password.
Check the zeroshell install for basic connectivity – use the Utilities page to hit various hosts on both sides of your network. Test some things by name, test some others by IP address. Do it while things are working, then do it when they aren’t. Compare.
Then, check hardware. If a NIC is going wonky, you might experience something like this.
That’ll give you a baseline of where to start looking.