I am interested in buying a website which has a great Alexa rank, but when checking on entourank.com, all of his scores are pretty bad.
Is it possible to have such a high trafficked website with no Facebook or twitter mentions?

I've never even heard of entourank, to be honest, and I put absolutely no faith in Alexa. Nowadays, I will check out Quantcast rankings for sites that have been Quantified, but other than that, I would recommend that you ask to see Google Analytics data before making a website purchase. Google Analytics will give you social activity now, so you should be able to get the complete picture.

What the analytics says.You can trust analytics data.