A dead-eye is a round block of wood with, usually, three holes drilled through the flat face. They are used in pairs as a crude block-and-tackle to apply tension to the shrouds of a mast. For a picture see HMS Pinafore. "Turning-in" refers to the art of wrapping a rope around the dead-eye and binding it with lighter cord. Fortunately, the plot line is in no way dependent upon your understanding all this esoteric nautical lore. Read on.

a better hand at turning-in a dead-eye
Act I
Sketch of a Deadeye