Recently one of our client's Joomla website was appeared to be blocked by Google. No search result is displayed at all even input the website's domain name.
You can try to input site:www.yourdomain.com in Google to see if your website is having an anti-bot problem.
We checked for a long while and finally found that a PHP program was using anti bot to disallow Google Spider to crawl the site. The plugin is named "E-Guard" and it has a option to block any bot outside of the website.
So if you meet similar problem, try to go to the backend and then extensions - plugin manager, see if E-Guard is there and then make sure the "Bot Access" is "yes". It was such a simple thing but did get us in trouble.