Apparently Final Fantasy XIV: A Realm Reborn is popular, a lot more popular than what even Square Enix expected, so much that the servers simply can't accommodate all the players pouring in, causing long waiting times and server congestion, and prompting the publisher to take drastic measures. Here's a message that just appeared on the relevant page of the North American Square Enix Store.

Due to the overwhelmingly positive response to FINAL FANTASY XIV: A Realm Reborn, we are currently experiencing extremely long wait times for users to be able to log in and play. As a temporary measure, we will halt sales of FINAL FANTASY XIV: A REALM REBORN’s digital download products so we can accommodate all of those wishing to play. We are working to expand our server capacity in the coming days.

We sincerely apologize for the inconvenience this causes.

The servers must be really at their breaking point if a publisher is basically refusing money (at least temporarily) in order to ease the congestion.

It's worth noting that if you don't want to wait and you still want to try and get into the game, the digital download version is still available at other retailers, like Amazon or GameStop.