The catch is that there are tolerance issues. Doron Swade's account of building the two existing Difference Engine #2 models (http://www.amazon.com/exec/obidos/ASIN/0670910201/donhosek) is a good example of where the challenges lie. It was just barely possible to do with 19th century technology. Physical mechanisms deviate from theory by quite a bit.
Antikythera mechanism was built before the Dark Ages, he just wanted to go back to the Dark Ages so he can do precision work by that time, he can probably build a battleship fire control system at that time.