Question: The word "dormitory" means "sleeping place". Is there any similar word for "eating place"?

Answer: 
There are lots of words for "eating place": dining room, cafeteria, restaurant, mess hall (military), galley (Navy), and so on.

Be aware that "dormitory" has a specific meaning, and is usually associated with schools, especially colleges. There are other words for sleeping place, including bedroom. If you're wondering if there are any words for eating place that are derived from the Latin meaning "eat" (as *dormitory* is from the Latin word for "sleep"), I'm not aware of any.
There is no single word for eating place, there are many.

* Restaurant
* Diner
* Cafeteria
* Eatery

You have to make a choice.