Jeff Bezos and Elon Musk, two of the richest men in the world, both with dreams of leading humanity out into the solar system, are fighting over the moon.

On Monday, Blue Origin, the rocket company founded by Mr. Bezos, who will step down as Amazon’s chief executive later this year, filed a 50-page protest with the federal Government Accountability Office, challenging a $2.9 billion contract to SpaceX from NASA to build a lander for American astronauts to return to the moon.