Many times a day, at different websites…hello Straight Dope…I get a screen freeze for a few minutes until a note comes up that the Server Stopped Responding’.
Am I being naive that someone or some machine is constantly monitoring the servers, and the problem gets fixed ASAP?
Or, in the real world, when someone finally sees the red light flashing, they get on it and wonder how long it’s been out?
We monitor our servers using other servers. The monitoring servers are watched in turn by another server. Let me tell you, keeping an infinite number of servers running is no picnic. 
There’s software that can do nothing more than simple “ping” tests to verify a server is on the network. There’s other techniques that can drill down through the many layers that make up a web application and check functionality from top-to-bottom.
In general, though, any professional shop is going to use some sort of monitoring software that alerts somebody to something going on. We have 24-hour operators on duty (operators are standing by!) that monitor automatically-created trouble tickets. Other shops might use pagers, etc. to alert to failures.
It’s not completely proactive but it’s better than waiting for a customer to complain.
That error message is generated by your browser. It means that at some point after you requested the page, the browser stopped recieving data from the web server for a certain amount of time, and gave up. The cause can be due to a number of factors, from Internet congestion to a temporary inability of the web server to keep up with demand.
There’s no need for a monitoring server in such a situation. Most of the time, such servers are not directly accessable by the public, rather, they send alerts directly to tech staff or automatically switch traffic from a downed server to backup ones.