I just stumbled on this thread, and wanted to say that I too, discovered (when I upgraded to iOS 8.1) that I could fix this "Server Not Responding" error by opening Settings/WiFi and since luckily we have more than one WiFi network here (to which I have access) I have been able to fix this problem by switching to a different WiFi network (and then switching back to the original WiFi if I needed that particular one for peripheral access).
I agree, this is ridiculous that Safari (and Mail, and other iOS apps that use internet access) cannot refresh the WiFi connection if it fails to connect while WiFi is active. That's what this is all about, even though other apps will say "No Internet Connection" while Safari gives this BS about it being all the "server's" fault. My iPhone or iPad Air 2 always have an active WiFi connection when this failure happens, and 90% of the time, it happens suddenly when I'd been accessing new data on the internet intensively, via Google search, or using an app like Etsy when between one breath and the next, it fails to remember it has been connected to the internet for hours. And yep, I did not move the location of my device, not more than an inch, anyway.