Server shutdown from power failure.
Mysql will not start now.
Disk is not full.
Syslog is below
Oct 11 15:03:31 joe mysqld_safe[24757]: started
Oct 1
getenforce
if it replies with Enforcing you have SELinux up and running. Temporarily deactivate it with setenforce 0 and see if MariaDB starts now! Rather common, especially on RHEL/CentOS/Fedora.
There's more about this further down, as well as in this official article.
There are more things in a UNIX environment that might prevent file access, than just user access rights.
Additionally there could be other unexpected factors, like ...
datadir being set to a place, where mysql doesn't have permissions (see /etc/my.cnf)Just to mention a view things off the top of my head (feel free to edit/add to this answer btw).
For a permanent solution, you could try to restore the appropriate security context, ...
restorecon -R /var/lib/mysql/
... or just deactivate SELinux (but think about this one a little bit before doing so), by editing the config (typically in /etc/selinux/config) and setting SELINUX=disabled as suggested in following article.
Obviously those are applicable to MySQL just the same way.