The latest version of Joomla is causing problems for web servers running older versions of PHP. Affected Joomla sites are still accessible, but users and administrators are unable to log in.
An announcement on the Joomla web site, and another in the Joomla documentation, provide details and workarounds for problems caused by the update, but web servers running PHP 5.3 won’t find them particularly helpful. If you administer a web server running PHP 5.3, the solution is to either wait for Joomla 3.6.2, or make some changes to a single Joomla file, as outlined in this fix on Github.
In case you’re wondering why any diligent web server administrator would still be running a version of PHP that is known to be insecure, what’s actually going on in most cases is that the admin is running a custom build of PHP that has had all relevant security fixes applied. For example, these custom builds of PHP are provided for Ubuntu LTS (Long Term Support) releases to allow for maximum security and stability.
Update 2016Aug05: That was fast. Joomla 3.6.2 is now available, and it fixes the PHP 5.3 compatibility issue.