baseball foul or fair?

Part of the batter's box is in fair territory, part of it is in foul territory. The foul lines (which are actually part of fair territory) run from the tip of home plate to first and third base, and the corner of the batter's box that is inside the diamond is fair territory.

If a batted ball stops in the box, fair or foul depends on where the ball stops.

I have this feeling that there's more to this question than you're actually asking.

Good question. Part of it is fair, part of it is foul. The baselines are supposed to extend into the batter's box, but they are never marked.

It neither foul or fair... or both, depending on how you want to look at it. The baseline runs from the backside of home plate to first and third respectively, with home plate being in fair territory just like every other base.

If the ball is in foul territory has not been touched by skill of every person or something and crosses into honest floor till now the backside and is in honest floor on the backside ir is a honest batted ball.