None of my ships are repairing??!!??

[Roanon]

And no, shipyards do not have to be idle to repair - it is in addition to their normal capacity. Similar to retrofitting where a single spaceyard is good for an unlimited number of retrofits, and does not interfere with normal shipyard functions.
But the amount of repair is limited, and per sector, not per ship, i.e. if you have several ships stacked only a few of them, sometimes even only one, will be repaired. Priorities are set under empire status (F11). Also, racial setup may influence the amount of repair available. All the repair capacity in a given sector is added up, then the percentage applied, then rounded.