IIRC, you can try again the next turn with a lowered chance of success, but you need to wait ~5 turns to get back to your original chance. There's a thread or something that discusses it ....
Edit: Eh.
http://www.civfanatics.com/civ3/strategy/espionage_missions.php
If a spy plant fails you will get a flag saying so. You cant plant a spy when flagged.
There is a chance of 1/3 for the flag to disappear each turn. Failing other spy mission, does not trigger the flag. You can plant a spy the same turn your spy has been exposed by a failed mission.
Not sure if there's any way to tell if the flag is up or down, though.