When Elon Musk agreed this summer to build a futuristic tunnel to Chicago's busiest airport on his own dime, many Chicagoans voiced doubt.

The idea sounded fanciful: Electric pods hurtling under the city at 150 miles per hour. And experts questioned whether Musk's company could dig a 17-mile tunnel in a few years.

Then Mayor Rahm Emanuel, the tunnel's main supporter, stunned Chicago this month by announcing that he would not run for a third term, casting more doubt over the project's future. A few days later, Mr. Musk, the chief executive of the electric-car maker Tesla, came under scrutiny for apparently smoking marijuana during an interview. The episode followed a summer of turmoil that led to the resignation of the company's chief accounting officer and raised questions over Mr. Musk's emotional state.