FFXIV isn't a cartridge-based game, so there is no built-in battery back-up. You'll need to purchase a separate memory card for saving data.

There's actually a neat easter egg during Gaius Baelsar's speech in Praetorium, where he'll make special comments if the game detects save files from other Square Enix titles. "Ah, I see you too enjoy Kingdom Hearts," that sort of thing.