Way to manually open doors if your Doors Subsystem is broken

Experiment Setup

I moved my whole crew up to the front of the Red-Tail, and evacuated the back half of the ship, leaving oxygen in only the three front rooms. This reduced my oxygen total to 20%. Then I closed doors and started timing every 10% increase.

Results

First I closed all the doors. Each 10% increase took 9.7, 10.5, 10.6, 10.2, 10.2, 10.6, 10.0, and 10.6 seconds. So it's very slightly less than 1% per second (1% per 1.03 seconds or 0.97% per second, possibly from 0.96% per 1 second which is a common math error when programming a game that runs at 60Hz).

Next, I closed only the outside doors, and opened all the inside ones. This took 66.3 seconds (8.2; 8.3; 8.3; 8.4; 8.2; 8.2; 8.3, 8.4). So oxygen does diffuse (faster?) through open doors. The background of the previously pressured areas turned slightly pink.

Next, I opened all the doors between depressurized areas but left closed the doors in the front which were still pressured. 82.6 seconds, basically the same as my first test.

Finally, I depressurized only the back three rooms of the ship, turned off my O₂ generator, closed the outer doors, and opened all the inner ones. The three back rooms re-pressurized despite my generator being turned off until they equalized with the rest of the ship, which was also steadily losing oxygen; eventually the whole ship depressurized.

Conclusions

This behavior is explained by a model where the O₂ generator works by supplying a fixed amount of oxygen to each room per second, oxygen is naturally consumed slower than this replenishment, and oxygen diffuses only between open doors. This means if a room with a high oxygen level has an open connection to a room with a low oxygen level it is giving oxygen to that room and receiving replenishing oxygen from the generator nearly as fast as it is giving it up and the other room is receiving oxygen from the generator, resulting in overall faster replenishment. If the doors are closed the new oxygen in the already-pressurized room is just "wasted".

Because the oxygen level required to keep your crew alive is fairly low it's often good to open doors to help re-pressurize areas faster assuming your oxygen generator is working.

FTL's model of oxygen seems to match our intuitions about how fixed volumes of gasses in an enclosed space work.

One interesting prediction of this model is that a sufficiently long path length of rooms may make it possible to survive indefinitely with a working generator despite a door open to hard vacuum. I've been testing this prediction on the Osprey for several minutes now, and it seems to be true.

If I open the second door on the top left, the room Mr Buga and Liam are in de-pressurizes. If I close it but leave the top one open, it re-pressurizes.

The only way to close your doors is to repair the "doors" system (appears as a vertical equals symbol). It only has to be repaired one point to allow you to close all doors. This will be very difficult to do in your situation as your crew may die of oxygen deprivation before they can make it to the door control and perform the necessary repairs. In this case you appear to be beyond all hope.