Microsoft has posted an announcement over on Steam that they are working to resolve the current ESO login issue and expect it to take days, not hours.
While MS figures this out, we on ESOC dev team were already able to find workarounds that let you get past the login and play.
First option is to utilize the latest ESOC Patch, where a fix is included and the log-in simply works. Most players are currently logging into ESO this way and playing on the patch. The ESOC patch itself also offers various other improvements over the regular one from 2009, such as a ton of new maps to play on, anti-cheating security measures, game-quality features such as two more possible teams, balance changes and other game bugfixes.
A little while after, we’ve also found a fix that lets you connect using any game version, by editing a certain configuration file of your operating system to essentially redirect how your computer sees the currently broken domain “eso.com”. This fix is detailed here.