We have received the attached and similar errors recently. Sometimes we can remedy this by restarting the computer and ensuring windows indexing is disabled for the C:\Users directory (although this setting does not seem to hold). This time, the "fix" isn't working.
We have an Amazon EC2 with ArcGIS Server 10.1 hosting an enterprise GDB. There is a feature service published from which authorized users connect to over https://, add to an ArcMap project, and create a local copy for editing.
The user that sent me this screen shot is following the workflow, but cannot successfully create a local copy for editing while others in the same and other office locations can.