Emblems can't be destroyed, as the "destroy" keyword only applies to permanents, and emblems are not permanents. R&D has no plans to create spells that are able to remove them. Emblems are a planeswalker-only thing.

Can emblems be targeted?

No, an emblem isn't a permanent. Emblems are put into the command zone, not the battlefield, and an object is only a permanent if it is on the battlefield. Oblivion Ring's ability et al can't target an emblem.

Do emblems disappear?

No, they do not disappear.

An effect that creates an emblem is written “[Player] gets an emblem with [ability]...” The emblem is both owned and controlled by that player. Therefore, the emblems do not disappear, because their owner(s) are still in the game.