The problem started to happen after I uninstalled GitHub Desktop from my computer, and it's still happening. This problem also happens if you try to clone a fork from a different user.
When I attempt to create a new GitHub branch from a repository I forked, the first image appears. The image says, "Unable to create" and "no such file or directory". The second image is the console output of the fetch command running, which says, "no such file or directory" and "failed to write object"
Try the same command from the command line, after installing the latest Git for Windows.
That Git might be more recent than the one embedded with GitHub Desktop, and should succeed in cloning the repo.