Barbetta has been in its Theater District townhouse for 100 years. I've never been, but my brother and his wife celebrated an anniversary there and told me the food was very good.
Rene Pujol, also in a townhouse, has been there for a little more than half that time (1950). We've dined there occasionally over the years, and the food has always been delicious.
Keens, which is on the outskirts of the Theater District, has been in its 36th St. location since 1885. Excellent food, good service, and incomparable old NY ambiance.