If you can borrow a second monitor to check if it exhibits the same problem, do it. Otherwise, I'd check your monitor's power management functions just to be sure it doesn't have it's own crazy brain telling it to turn off. (It's a long-shot, really.) If you're sure it's not your PC telling the monitor to take a nap or, the monitor dozing of it's own volition, you probably have a hardware issue and it's time to retire the old CRT. There may be safety circuit in the tube to shut it down if it senses a problem like excessive current draw.
Before you panic, first, do what you can to elliminate your PC as the cause. You can also borrow a laptop or, something, to test out your monitor.