There is a base, or a ship, or a city, where our heroes spend their time throughout the story. It may either be the location where most of the story takes place, or it could be a set piece only seen once in a while. Through some course of events, be it an enemy attack, or a natural disaster, or some kind of accident, everything has pitched into a state of chaos. People are wounded, equipment damaged, chaos and confusion reigns supreme, and everything is on fire.
The first order of the day, aside possibly from dealing with the cause of the damage, is putting out the fires, tending to the wounded, and repairing the damage so everything can go back to normal. Expect all this to be going on even as the driving crisis that caused everything is still ongoing. The main characters will often be split apart and unable to communicate with each other or rely upon each other directly. Expect at least one character to reach down inside himself and find the resolve and resourcefulness he needs to solve a major problem without the team mates he normally relies on. The Engineer and the Old Soldier will be at their best here, leading their men in the dangerous and critical work of getting everything operational again.
The trope draws its name from an expression used on naval and merchant vessels, for urgent repair work that is done while the ship is at sea. If unsuccessful, our characters will often be forced to segue directly into Abandon Ship.
Note: when applying this trope to video games, it should only be done when dealing with such repairs as a plot point, as opposed to game mechanics centering on unit healing or repair.
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In Star Trek II: The Wrath of Khan, the Enterprise is taken by surprise in the initial attack, but Kirk and Spock are able to turn the tables and deal some swift damage to Reliant. Both ships are forced to withdraw and effect repairs before they can fight again. In the final battle, both ships are again crippled, and it is only a Heroic Sacrifice by Mr. Spock that allows them to survive.
In Star Trek IV: The Voyage Home when Kirk asks when they can get their captured Bird-of-Prey under way, Scotty quips, "Damage control is easy; reading Klingon, that's hard."
Again, during the battle in Star Trek: Generations, Geordi LaForge is seen ordering his repair teams around, although it ultimately turns out to be futile. Interestingly, this scene is a line-for-line recreation of a similar scene in the series epiosde "Yesterday's Enterprise", which nicely foreshadows the Enterprise's fate.
In Star Trek: Insurrection the Enterprise gets into another space fight. It's not really seen, but right before it starts, LaForge leaves the bridge for Engineering, knowing what he's going to be doing in short order.
The Star Trek II scene is mirrored in Star Trek Into Darkness, this time the roles are reversed. Spock is commanding the battle and Kirk is the one who does the Heroic Sacrifice to repair the ship before it crashes or burns up in Earth's atmosphere.
A villainous example in Iron Eagle, due to Chappie and Doug hitting the airfield first, and Colonel Nakesh gives his men one hour to reopen the airfield.
The Phantom Menace: After Trade Federation battleships shoot out the shield generators on the royal cruiser, R2-D2 and several other astromechs are called out to fix it. R2 gets it done but all the other droids are blown away in the process.
The Empire Strikes Back: Han and Chewbacca are shown trying to get a malfunctioning hyperdrive online while being pounded by Imperial Star Destroyers.
In Down Periscope, the Stingray starts bursting water pipes while sailing dangerously close to a freighter to sneak into a harbor, forcing the crew to patch the lines before they cause additional damage, leading to Stepanek'sBig Damn Heroes moment.
Red Storm Rising had a number of instances, notably the damaged USS Nimitz limping back into port after being struck by missiles, a soviet freighter carrying troops and equipment having to fight fires after being strafed by F-15s, the seawater used to put them out inadvertently ruining a lot of missiles that had planned to install in their new base. It isn't seen, but it's obvious that this is what's going on aboard the ill-fated USS Providence. Finally, the Soviets manage to take over a USAF base in Iceland, and when the Americans attack trying to render it useless, cratering the runways, the Soviets use the repair materials the Americans themselves had left in the event of such an occasion and have them back in use before nightfall.
Happens fairly often in the Honor Harrington series, particularly in the books that center on one or two ships. Books which focus on fleet actions will mention the damage control in passing, rather than focusing on it in depth.
On Basilisk Station shows some of the issues that a crew of a ship in battle have to deal with. When the missile launchers on one side of the ship are damaged, along with the automated systems for transferring missiles from one magazine to another, crewmembers have to move the missiles by hand with the assistance of anti-gravity rigs, which make the missile weightless, but do not eliminate its inertia, causing one crewmember to be crushed (nonfatally) against the wall when the ship takes a hit.
In the Warrior Cats series, a forest fire burns through ThunderClan's territory in the fourth book. In addition to killing several cats and driving out all the prey, the camp itself was destroyed. They have to try and rebuild it with whatever little they have left to work with, and try to get back to a normal lifestyle, before the other Clans take advantage of their vulnerability.
The Argo II in The Mark of Athena takes frequent damage through the course of the book, but since they are attacked almost every time they set down they have to do a lot of repairs on the fly, in one case their means of escape does almost as much damage as the attack, meaning Leo has to do some fast work just to keep the ship from flying apart. Another time after being attacked by a sea monster Leo got washed overboard so the others, under Annabeth's direction, had to do enough repairs to keep them from sinking.
Battlestar Galactica (Classic) episode "Fire in Space". Cylon raiders ram the Galactica in suicide attacks, causing fires to break out throughout the ship. Several of the crew are trapped and Commander Adama is injured.
A subtle example: Part of the station (one of the fork-like protrutions on the top of the station for handling cargo) is blown off during a battle with an attacking ship in the season 2 finale. In the season 3 intro, the damaged section can be seen re-attached and surrounded by scaffolding.
The season 4 episode Endgame has a fleet of ships get crippled due to the heroes' sabotage just before the final battle of the season. The repairwork isn't shown, but is discussed soon after, and later referenced in passing when one of these ships arrives during the final battle just in time to save the Aggamemnon.
"Engineering, this is General Lefcourt. In case you didn't notice, the enemy just pulled our shorts up over our head and tied them into a knot. You will get the ship under control ASAP, or I will come down there and skin the hide off every last one of you."
In the first episode of Last Resort the crew of the USS Colorado are shown repairing the sub after a missile attack.
Another first episode, this time for seaQuest DSV, had the titular sub taking a direct from a torpedo. Afterwards, the first officer is talking about the repairs being made. Later a more extensive effort is shown tryign to root out a virus that's been screwing up their computer systems.
Firefly: In "Out of Gas", Serenity is crippled by an explosion which disabled their life support. In this case, the required repairs are implied to be simple enough, but they don't have the required part. Mal sends them off while he waits on the ship for someone to answer their distress call.
The card game Red November is nothing but damage control on a submarine prey to fires, flooded compartments, and the looming threat of a Kraken attack.