What counts as "Without a Weapon"?

The best strategy to go about killing Deathclaws without a weapons would be to equip your Dwellers that are fighting with the best armor you have. The mission to take them out is to take them out completely WITHOUT using a single weapon, which means, yes, you will have to smack 'em down without using your trusty alien gun.

I would suggest limiting your population down to exactly 59 dwellers before you are prepared to fight Deathclaws. Just as ethanflips said, this task is annoying, but you'll get through it. If you simply want to forget about it, don't forget that you can skip objectives once per day. Hope this helped and answered your full and complete question.

You would need to put very good armor on the dwellers, so they don't die quickly and keep healing them. If anyone in a room has a weapon where a deathclaw is it won't count. That's how I did it. It may take more than one try which can be annoying, but it's not too hard...