Can the warforged be woken during its 6 hour long rest. Or is it inert for that time?

The warforged is awake at all times, with the possible exception of being reduced to zero hit points.

This means that they are simply statue still during those 6 hours, until/unless something requires them to act.

From Sentry's Rest:

"In this state, you appear inert, but it doesn’t render you unconscious, and you can see and hear as normal."

Also Relevant, from Warforged Resilience:

"You don’t need to sleep and don’t suffer the effects of exhaustion due to lack of rest, and magic can’t put you to sleep."

A warforged can actually keep watch while resting.

@DxJxC Indeed, which kind of makes feats for alertness and perception really good for them.