I got close to losing:
My sole boarder could not attack this enemy crew member or move to the free space, I think, and the weapons kept shooting. However, after some time (30+ seconds?), the enemy crew member moved elsewhere (maybe my sole rocket caused damage to the enemy ship?), and got killed by my boarder. In FTL, boarders that are in a room without crew but with a ship system, attack and damage that system, meaning that the defending crew have a reason to go to that ship system beyond just fighting, namely in order to defend the system and try to prevent it from being damaged.