Four private astronauts aboard an ambitious space mission led by a billionaire entrepreneur traveled farther from Earth on Tuesday than any other human being in more than half a century.

Two of them, Sarah Gillis and Anna Menon, have now gone farther from the planet than any other women ever.

The mission, named Polaris Dawn, lifted off through a break of favorable weather before sunrise on Tuesday. The flight had been grounded for nearly two weeks by unsettled weather in and around Florida.

The astronauts, flying an elliptical path around Earth in a SpaceX Crew Dragon capsule, looped outward as far out as 755 miles above the planet’s surface. The mission’s orbits were carefully planned to reduce the hit of radiation the crew would absorb, and to minimize the chances of being struck by tiny bits of rock crisscrossing the solar system.

The journey on Tuesday was only a small fraction of the nearly quarter million miles that NASA’s Apollo astronauts traveled to the moon. But after the last mission going there in 1972, humanity has stayed close to our planet, not venturing beyond orbits a few hundred miles up.

The Polaris Dawn mission, led by Jared Isaacman, founder of the payment services company Shift4, is a collaboration with SpaceX, the rocket company founded by Elon Musk. It is the first of three missions designed to spur technological advances needed for Mr. Musk’s ambition to send people to Mars eventually.