Apple recently released iPod classic Firmware 1.1.1, which appears to fix a problem where an audible hiss was apparent when the unit was turned off.
Some users have speculated that the hiss was a byproduct of a feature that allowed the classic to wake from sleep upon insertion of headphones; that feature no longer works in 1.1.1.
Other users have complained of audio still being audible even when the volume is set to zero (an issue that pre-dates this update); Apple has a support document on the issue, which says “If you want the iPod audio to be silent, press the Play/Pause button to pause playback.” iPod classic Firmware 1.1.1 is available now through the update feature in iTunes.