How do I fix this broken stasis pod?

Solution 1:

There's an event where some Zoltan scientists will ask if they can run some tests on your ship and crew. If you have this pod, when you meet with these scientists, you can instead choose to allow them to inspect the pod.

From the pod:

You'll gain a new crewmember, who is of a race that preceeded the Rock people. He will ask you to embark on a quest near the Rock homeworld. He is sort of a cross between Rock and Human, and has a "Lockdown" ability that is unique.

Once you've completed this quest stage, you'll get another for the Rock Homeworld sector. If you've already passed this or don't have it on your route, you're out of luck. If you manage to finish this stage of the quest:

You will unlock the "unidentified cruiser" - the Crystalline Cruiser in the ship hangar.