The Royal Navy said the exercise on a surface ship triggered a fire suppression system.
Emergency services were called to the naval base just after 5:30am.
The staff members were taken to the Royal Alexandria Hospital as a precaution and their families have been informed.
An investigation has also been launched.
The Faslane base on the Clyde is home to the UK’s fleet of Trident nuclear submarines.
A Royal Navy spokesman said: “We can confirm that there has been an incident on a surface ship alongside HM Naval Base Clyde this morning.
“A routine fire exercise triggered the fire suppression system, which resulted in a number of personnel inhaling a gas.
“Six personnel were taken to hospital as a precaution; all others were declared fit at the naval base’s sick bay.”