Do any errors pop up on your site if you head to your site’s wp-config.php file and change this line of code:
define('WP_DEBUG', false);
to this:
define('WP_DEBUG', true);
That will display PHP errors if any exist, which might be quite useful here.
So make this change to that file, then try activating The Events Calendar again. If any errors do pop up, please copy the full text of the errors and paste it here.
I *fixed* this by adding the code below to my wp-config.php file. The plugin activated and the front end worked. It also fixed another issue I was having with my theme.
define('WP_MEMORY_LIMIT', '256M');
The site was recently migrated to new hardware by Verio, so that would explain the change.