It is possible for an object to be in the shadow of Earth permanently - no special restrictions on the Earth or Sun needed.  There is a special place called the L2 Lagrange point that allows an object to orbit around the Earth at about the same rate as the Earth orbits around the Sun - thus always remaining in shadow.  However, L2 is not a super stable point to orbit in, and thus objects tend to fall away.

...more info to come.