Does a power strip use any power in the off position?

If the power strip has an indicator light it uses a very minimal amount of power, especially if it's a neon or LED indicator. If the power strip uses either of those, the power use is effectively unmeasurable. Incandescent indicator lights use a very small amount of power.

But in reality, indicator lights are off when the power is off, so actually power strips use no power when they are off.