I'm trying to figure out how a user has got himself in this predicament. Last month he had Windows 7 go bad on his PC, so we reinstalled windows, drivers, and apps. He reinstalled Desktop 10.2.2 from media I had downloaded, don't know which patches he applied. There was a partial backup of some of his data and settings, so some of the maps and data he had on C: were lost. But many of his projects and much of his GIS data were on an external USB drive, so I thought at least he wouldn't have trouble with that source. However, some of those mxd's don't completely load. Looking at the data sets that will not draw in the Catalog window, we see that they are listed, but have a blue ? in front of them, not the yellow !. All the ones like this that I saw are shapefiles. They don't draw in ArcMap or preview in catalog. What does the blue question mark mean?
I suggested that he point Windows Explorer at the top folder of his GIS data on the USB drive and try a reset of owner to be his new login to everything in the tree, even though his new log in appears to be the same as his old log in.
There are other links that might be a bit more modern
27429 - Recover a corrupt shapefile
arcgis desktop - Fixing corrupt shapefile? - Geographic Information Systems Stack Exchange
As Darren mentioned, they may be beyond hope, but not bad to have a list of tools/resources.
Convince yourself that this is corrupt by trying to open the .prj file in a text editor. It should be well-known text. It isn't.
Yes, the prj files of the bad shapes open like they are binary, the prj files from good files open as readable text.
Poor guy, the whole USB drive is a mix of some good shapefiles and geodatabases, but a lot of bad ones. He thought this USB drive was his reliable backup. It's a WD My Book.