A curious addendum to Professor Moose's excellent explanation is that it appears prior to the Calamity occurring, Bahamut's existence was in fact known on some level, whether through ancient stories or cultural memories or through some other means by the general public - in 1.0 the first Job quest for PLD featured this curious exclamation from Captain Jenlyns, the captain of the Sultansworn of Ul'dah:



Even more curious, is the line was retained for the exact same quest in ARR as well. So, how did Jenlyns know of Bahamut? Foreshadowing not only what was really inside Dalamud, recalling cultural memes of the real Bahamut killed by the Allagans in Meracydia? It always puzzled me why Jenlyns said that, although it could have just been a sly bit of hinting how the Calamity was going to unfold by Ferne.