How did Apollo 13 get home after an oxygen tank exploded?

By turning the lunar lander into a lifeboat and taking the long way home around the Moon. On 11 April 1970, Apollo 13 launched to land on the Moon. Two days in, a routine stir of an oxygen tank ignited damaged wire insulation inside it, venting the contents of both service-module oxygen tanks into space. The landing was aborted, and the crew had to get back alive.

The service module was dead, so the crew moved into the lunar module. It had charged batteries and full oxygen tanks, so Flight Director Gene Kranz had the crew power it up and use it as a "lifeboat", a scenario anticipated but considered unlikely. It was designed to support two men for two days on the Moon; Mission Control improvised new procedures so it could support three men for four days.

Getting home meant a choice. A direct abort would have used the service module's main engine, which the accident could have damaged, so Kranz chose a longer route: swing around the Moon and head back to Earth. The crew endured cold, down to 3 °C in the darkened spacecraft, and an improvised carbon dioxide filter built with duct tape and plastic covers from manuals.

It worked. The crew returned safely on 17 April, and Odyssey splashed down in the South Pacific Ocean. The cause was damaged insulation on a tank whose heater tube may have run far hotter during testing than its gauge could show.