One of the most challenging projects in the world is being attempted beneath one of its most densely packed cities.

If it works, Mumbai will become the planet’s most crowded metropolis to build an underground subway. More than 8,000 workers and a fleet of 360-foot-long boring machines are working 24 hours a day—even through monsoon rains—to finish the 27-station, 21-mile subway through some of the world’s most densely populated neighborhoods, around the edge of one of Asia’s biggest slums, below an airport and under temples and colonial buildings to ...