A first encounter with the Lunar Barracks can catch players off-guard, resulting in defeat, or reloading a save, so lets prepare best we can.

-Defences- A sturdy gate and some archers are in order, to defeat the Barracks. As soon as the structure spawns, enemies will be pouring out of it (endlessly), so we must hold them back as long as we defeat it. (Note that the usual nightly wave will also be coming at us.)

-Melee units- The Lunar Barracks is resistant to arrows (like enemy shield troops), therefore we must use melee units to bring it down. Any type of melee unit is useable, as long as we have the sufficent numbers. 20 or more units should do the trick, make sure to upgrade their gear, and give them some banners, that can heal them.

-Archer support- The enemies pouring out of the barracks will try to kill your melee units, so we must defend them with archers, until they do the demolishing. 10 or more archers are enough to protect our troops, position them further away from the Barracks, so they wont be priorized as targets.

Now that we know how to prepare, lets discuss how to spawn the Lunar Barracks. The Barracks are spawned by destroying a specific Lunar Tower, bringing your cursor to the Tower will tell you what it will spawn, upon destruction. The Lunar Barracks will be spawned on the night of the Towers destruction.

To ensure that in the night our units will be harmed the least, position our melee troops and archers behind the enemy spawn(s). You can see enemy movement paths by bringing your cursor on the day/night cycle menu/counter. This is best done in the day, so that stray units or enemy archers wont pick on our troops.
Upon the spawn on the Lunar Barracks wait a bit, so that the bulk of the enemy force leaves the area of the structure, so that we deal with the least amount of resistance. After the bulk leaves, send our melee to the party, and make sure to back them up with our archers, but from a safe distance.

If we done everything right, the barracks should be down, with minimal or no losses on our end.
Congratulations! We just unlocked and achivement, and counquered a great challenge!