Shoki is right - there actually are Phoenix Downs in FFXIV - they're classed as unique items though, that you can only use on a KO'd character outside of battle (you can only hold one at a time in your inventory, but by hiring two retainers you can keep two more - one on each). At the moment the player can only aquire three as optional rewards from three specific sidequests, the first from Quarrymill, the second from Camp Skull Valley and the third from the Ceruleum Processing Plant.

Other than that, the only way you can revive a KO'd player is to equip the CNJ spell Raise as a cross class ability which is learned at level 12 (ACN also learns a similar spell called Resurrection, but that can't be used cross-class).