Any users not allowed to export data, should probably not be using ArcGIS for Desktop at all... there is just to many access points to data and tools for data conversion.
Instead, they probably should be using some custom web-application to access non-editable ESRI Image or OGC WMS services serving out just images and not true features. By the way: you can access these webservices too through the ArcMap interface. This might be a part solution: make the Feature Classes only directly accessible to the geodatabase administrator, and than create ESRI Map / Image or WMS webservices using ArcGIS for Server to allow access to other users from within ArcMap.
If the users do need editing capabilities, than a custom web editing application using ESRI Feature Services or OGC WFS-T (transactional WFS) might be in order, but that would still allow access to feature geometry data somehow, but at least not directly from an interface like ArcGIS for Desktop with its many options...