What is the specific requirement for achievements like "Alone Tonight"?

I have not attempted this myself, but people are reporting that you are not allowed to have any set bonus from sets listed.

That means you can wear at most 1 piece from each of those 24 sets (there are some very useful single pieces there, like those that give +45-60% to skill damage).
All other sets are fair game - Chantodo's, Bul-Kathos', Bastions of Will, Endless Walk, crafted sets.

There is also a bug that allows to grab those Achievements and a Conquest easily. Set items do not provide bonuses to a lifeless corpse, so you just need to die right before your damage over time effects to finish off the Rift Guardian.