Hell - my next-door neighbors have horses in their backyards and so do three houses directly across the street, and I've never seen one of these. Everyone has 150s or 250s.

You probably have-at construction sites or something, with giant beds meant for towing huge digging equipment, and so never noticed them at all.

I saw one once on T Pains Cribs episode

You almost certainly have seen a F650, you just probably didn't recognize it because Ford only sells them incomplete (without the bed) and they are highly customized to fit their purpose. Commonly they are dump trucks, bucket trucks, heavy wreckers (tow trucks for transports), box trucks, etc. Here's some good pictures of what they can be used for.

I've seen maybe three of them, and one was at the Chicago Auto Show.

Most people in the market for a truck like this go for the dedicated commercial manufacturers, like Mack or International.