This has been happening consistently for the last two weeks. Whenever I restart my computer, or go a few hours without logging into FFXIV, I get the "System Error has occurred: -24723138982" or whatever, then ten minutes later I'll get the "Unable to complete version check" then ten minutes later I'll be able to log in.

Here is a fix that has worked 100% of the time for me, as this is an issue with the server and your computer's DNS:
  1. Press Windows Key and R at the same time
  2. Type cmd and press enter
  3. Type ipconfig /release and press enter
  4. Type ipconfig /renew and press enter...wait 5 seconds for it to complete
  5. Type ipconfig /flushdns and press enter
  6. Try opening FFXIV again.

I hope this works for other people, as it is an incredibly annoying bug.