I’m going with Gemini 8, a flight that came within seconds of killing Neil Armstrong and David Scott.
Gemini 8 was docked with the Agena booster, which was a “docking target” for this mission. The plan was for Scott to perform an EVA after the docking as practice for later missions.
Command Pilot Armstrong had docked the Gemini capsule to the Agena booster (which had been put in orbit weeks ahead of the mission) flawlessly.
(Agena booster – NASA image)
After docking, and before the start of the EVA, Armstrong noticed a sudden change in the attitude of the combined craft. He took control to stop the movement, but as soon as he ceased thrusting, the “roll” movement started up again.
The combined Gemini capsule and Agena booster started rotating (end for end), and Armstrong trying to keep up and null the motion back to zero. His initial thought was the maneuvering system on the Agena booster had malfunctioned, so while he tried to control the tumble, Dave Scott hit the “undock” switch. The Gemini capsule spun free. Armstrong then noticed that the “on orbit maneuvering system”, OAMS, fuel was still dwindling rapidly on the Gemini capsule, making the emergency even more dire – that system was required to de-orbit and come back to earth!
双子座机舱和Agena助推器的组合装置开始旋转（首尾相连），而阿姆斯特朗则尝试保持该速度并想将其减小到0。 他最初认为是Agena助推器上的操纵系统发生故障，因此当他试图控制翻转时，Dave Scott按下了“不对接”开关。 双子座机舱自由旋转。 然后，阿姆斯特朗（Armstrong）注意到，“在轨机动系统”（OAMS）的燃料仍然在双子座太空舱上迅速减少，这使得情况变得更加严峻——该系统需要脱离轨道运行并重返地面！
At the maximum rotation, the Gemini Capsule was moving at 296 degrees a second! This was very nearly enough to render the astronauts unconscious.
Armstrong shut down the OAMS system, finally understanding that it was a stuck thruster, and used the De-Orbit system to null their motion to zero (saving their lives). Power the OAMS system back up one thruster at a time, they managed to isolate which thruster was stuck “on” and shut it down… They barely had 25% of the OAMS fuel left.
NASA had Gemini 8 reenter after one more orbit, three days before the recovery ship was supposed to be on-station to recover them.
(Gemini 8 during recovery – NASA image)
Had Neil Armstrong lost consciousness from the rapid tumble, or been unable to figure out which thruster was stuck “on” in time to keep enough fuel to de-orbit, history would be considerably different.