Back from the brink
You first need to get your reactor damaged enough for it to go subcritical (once you lose the 3 green bars it goes subcritical and the green reactor bar is replaced by a red “subcritical” one – once that depletes the reactor goes boom and you go dead).
Getting it subcritical in a controlled manner is… well, good luck with that. From what I experienced it’s up to the AI to be competent enough to actually hit your reactor efficiently/consistently without your shields preventing this.
This means that there are two possible options:
1. Shields down and hope they don’t retreat before they pummel your reactor. To increase your chances, get in a 1v50 battle. Keep in mind that they’ll hit other things besides the reactor. While systems and engines can be repaired ad infinitum, your healing options are finite. If you get lucky and they damage the reactor send your engineer to fix it ASAP. You won’t have forever to fix it once it goes subcritical.
2. Hope you get boarded and let the boarders smash the reactor. Sounds like a better choice however, there’s no guarantee they’ll go straight for the reactor. If they only damage another system, no problem. If they attack your crew (and they will…) there only so much punishment you can take before you have to respond. If you get lucky and they damage the reactor, get other crew members in the fight while sending your engineer to fix the reactor. You won’t have forever to fix it once it goes subcritical.
Chances are that if you’re new to Space Crew/Bomber Crew you’ll get a subcritical reactor without even trying to. If you know what you’re doing, this will actually be the hardest achievement to get.