Final answer:
Apollo 13 encountered an oxygen tank explosion in the service module shortly after launch, resulting in the mission being aborted.
Step-by-step explanation:
The problem that Apollo 13 encountered shortly after launch was Option 4: Oxygen tank explosion in the service module.
This explosion occurred during the second stage of launch on April 13, 1970, resulting in the mission being aborted and the crew narrowly escaping with their lives.