Kōbe's Chinatown – Nankin comes from Nanjing; machi just means town – dates to the early days of the city opening its port to foreign traders. It was rebuilt after the 1995 earthquake and has all the visual signifiers of Chinatowns the world over: tiered gates at the cardinal entrances (except for the north side, guarded by lions) and lots of restaurants.
It's definitely touristy, but it's fun: most restaurants have stalls out the front selling street food, like nikuman (steamed buns, usually filled with pork; baozi) and chimaki (sticky rice wrapped in bamboo leaves, also often filled with pork; they're also called zongzi) for a few hundred yen each.