If there are no Archfiends on your side of the field, "Falling Down" is destroyed immediately, even in the middle of a chain.
The Archfiend you need for "Falling Down" doesn't have to be a monster. Cards like "Axe of Despair", or an Archfiend monster equipped to your "Relinquished", will prevent "Falling Down" from being destroyed.
You can activate "Falling Down" when there are no Archfiends on your side of the field, but it is destroyed immediately before it is equipped to a monster.
You cannot use "Barrel Behind the Door" against the damage from "Falling Down" because you cannot use "Barrel Behind the Door" against a Continuous or Equip Spell Card.