I'm a sporadic shortwave listener and I've only ever come across two number stations in my years of listening. Here is one from my logs:
Date: 7/19/1999
Time: 3:05-3:10 UTC
Frequency/mode: 13430 kHz AM
Signal/Interference/Overall (SIO: 1-5, 5 best best): 433
Description: Female announcer with a Russian accent reading sequences of five (?) numbers.
The other one was a Spanish-speaking woman reading numbers for about the same duration. Most people live believe that they are messages broadcasted to spies in the field who have a temporary decryption key for the message. Some are even known to play jingles before the number sequences are announced, such as the "Lincolnshire Poacher" station (which, not surprisingly, plays the song of its station's given namesake.)
Spynumbers.com has a pretty comprehensive database of loggings and patterns attributed to certain stations.
And MrSelfDestruct, I wouldn't say "many" are 24/7. Some tend to broadcast unmodulated carrier signals for a while, but the actual numbers reading are quite short. Either way, it is still an interesting phenomenon considering how many other, more seemingly safe ways there are to transmit messages. By the way, for those who are interested in numbers stations, I also suggest tuning into shortwave pirate radio. Transmissions are usually in the 41 meter range (6950 kHz frequently), and the content can be surprisingly good in spite of the audio quality.