How do I unscrew a hex screw that doesn't have any grip anymore?
You may have some luck with a screw extractor. They come in various styles - here are two of them:





The extractors are designed to screw themselves down into the head of the screw while at the same time applying torque on the screw in the direction that would loosen the screw. Your described screw had a hex socket type hole so the extractor may be ready to try. Sometimes it is necessary to drill a small hole in the screw head to engage the extractor. The first type shown has extractor on one and a suitably sized drill on the other end. 

It the extractor fails to remove the screw or if the thing is too far gone already to accept a screw extractor then it would likely be a next step to remove the core of the screw by drilling it out. With care you can drill down the center of the core and use a diameter that is smaller than the outside thread diameter. Then you can come back with a threading tap to remove the reset of the screw material and clean up the threads in the mating part. Here is a picture of a typical threading tap.