Eyes have different innervation than most parts of our body. Most of the body is controlled via the spinal cord, but the structures in the head, like eyes, are controlled via cranial nerves that come directly from the brain. There are 12 pairs of cranial nerves that are traditionally numbered from top down. Eye movement nerves are number 3, 4 and 6. If you got your connections disrupted on the level of, say, number 4, you still have some movement left in the eyes/eyelids, but can't move anything else in your face. The whole body is paralyzed, too, because the spinal cord / brain connection is off.

Some people are/were totally paralyzed, yep. There is some ongoing research on recognizing and helping such people. Doctors can scan for brain activity and use machines that interpret this activity as attempts for communication. They've actually made a brain implant that allowed a patient to ask for beer, curry and some music. It's amazing.