SHARMA, Saudi Arabia—This seaside corner of northwest Saudi Arabia is so barren that the only abundant resources a group of consultants could identify were sunlight and “unlimited access to salt water.”

But Saudi Crown Prince Mohammed bin Salman didn’t see a wasteland when he landed in his helicopter here a few years ago. He saw the future— and hatched a plan for a $500 billion city-state to cover 10,000 square miles of rocky desert and empty coastline to attract the “world’s greatest minds and best talents” to the world’s...