Three years ago, a billionaire entrepreneur named Jared Isaacman made a groundbreaking trip to space. That spaceflight, which Mr. Isaacman called Inspiration4, was the first to orbit the Earth without a professional astronaut aboard.

Next week, Mr. Isaacman, the founder and chief executive of Shift4, a payment processing company, is scheduled to head into space again. This time the itinerary is longer, more daring and riskier, and includes a spacewalk, the first by private astronauts.

The mission, named Polaris Dawn, hearkens back to the earliest era of spaceflight, the 1960s, when pioneers like Yuri Gagarin of the Soviet Union and John Glenn of NASA pushed the boundaries of what had been accomplished in space, learning how to survive and…