Atomicrops

You will need to get lucky and/or just keep playing to find the Megaton Tractor. Once you find it you'll need to land three hits on the moon during the blue moon phase. If you can get extra charges great. To be clear it's the big blue moon ball that chases you. Additionally, since I dont see it mentioned you can actually return to your farm to stop it chasing you and go back to where it was to land more hits and still get the achievement. The hits seem to stack (assuming you do it all in one go).

Unfortunately I believe that this is a luck based achievement. You will need the Megaton Tractor in order to kill the moon and as far as I know, the Megaton Tractor can not be bought from the store; it has to be found in one of the biomes. Other than that, this achievement is pretty simple. Simply make sure that your tractor has at least 3 charges on it. (I would try and get more in case you miss the tractor drop). But when the blue moon that chases you around outside of you farm (the one that tells you to go back to your farm) drop the Megaton Tractor on it. If memory serves me right it take 3 drops of the tractor to kill the moon. I personally didn't see any increase in damage to the moon with a gun, so I don't think you can shoot it dead.

Year Recommendation: 1
Playing this on year 1 will give you the best opportunity to get money and good guns to be able to explore the other biomes and find the tractor.

Edit: I got this achievement a while after the Feline Good update for consoles.

Was playing on year VIII, working towards year X, wasn't going for the achievement, but randomly got a megaton tractor from one of the tier I tree guardians. I explored some more of the biomes hoping I'd find some yellow mushrooms to stack up some tractor charges, but I found none, maybe this was because I was holding a megaton, or I was just really unlucky, because usually I find quite a few of them.

Had a fully upgraded SMG, while clearing the jungle biome tier II, then I got an evil moon to spawn, I'm calling it evil, because it wasn't the one that let's you relax, but the one with the witches and aliens.

Immediately I used my megaton tractor on it, but this wasn't enough to take it down, I started spraying it with my smg, this actually slowed his pace, making the moon hardly move at all, I started to kite the moon towards places that still had enemies to recharge my tractor, while also frequently firing at the moon to slow it down, I have no clue if it mattered, it died not on the second, but the third hit with my megaton tractor and the achievement unlocked immediately, I died soon after xD.

This is how I got it, but it would probably be easier to do it on year I. I wouldn't focus on this achievement as you can't control if you will get both a megaton and a moon to spawn in a run, but when you do find a megaton, hold on to it and pray to the rng-gods.