World of Warcraft Classic is a cesspool of bugs, errors, and glitches—error code 51900101 being a prime example.
The error code tends to crop up in the Classic and Burning Crusade Classic versions of the game, abruptly disconnecting you from the servers, and preventing you from logging back in and playing the game.
The fastest and most reliable way to solve this issue is to delete your WoW cache folder. Doing this will reset your user interface, keybindings, and addons.
If you want to preserve your settings, consider taking a few screengrabs just to make sure you don’t lose anything of importance. It can be a pain to go through all your settings again and find the right settings.
When you’re ready, head to your World of Warcraft folder, choose _classic_, right-click the cache folder, and click delete or manually move the file to the trash. Once it’s deleted, a fresh cache folder will immediately pop up and replace the old.
If this method doesn’t work, the problem could stem from your internet service provider or a firewall, which could be blocking access to your game. If this is the case, try troubleshooting your internet and temporarily disabling your firewall.