According to the H&K Sports Fields (commercial) website (refer to the link, below), the batter's box starts 6 inches from home plate, and measures 4 feet long by 6 feet wide, total (i.e., both the right and left sides combined).
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The center of the batters box is even with the center of home plate. since the plate is 17" long the this means the front edge of the batters box is 27 1/2" in front of home plate and the back edge of the box is 27 1/2" behind home plate.
The front of the batters box is three feet from the middle of home plate which is where the angles start that lead to the point facing the catcher.
90 Feet from the plate in Major League Baseball. 86-94 feet from the batter's boxes (depending on the side).