CRT screens simply don't emit light when it's black. Thus, the contrast ratio is theoretically infinite.
However, CRT screens have glare problems, so if you're using any sort of lighting in your room, the "real" contrast ratio goes down really quick. If I remember right, it becomes around 20:1 to 30:1 in normal lighting. I'm not sure about CRTs with anti-glare coatings though. LCDs also suffer from the same problem (the only time they're anywhere near their advertised ratios is if you're in a dark room) but because the surface is inherently anti-glare, the ratio is around 100:1 or so in normal lighting.
Where LCDs fail at is with regards to near-black screens. They simply suck with them, and backlight bleeding issues on certain monitors don't help. Unfortunately, game developers currently like to use really dark levels and stuff. For some reason they think using just 10% of the brightness spectrum (regardless of CRT or LCD) makes for a better game. At any rate, aside from response time, this means that the contrast ratio should be a big concern for gamers.