The Rusty Key is used to enter The Sewers. It is obtained from Gunther in a cutscene after donating 60 items to the Museum. The morning after the 60th item is donated, he will greet you at your front door, thank you for your contributions to the museum, and give you the key.
Its existence is first referenced by Jas and Vincent in a cutscene in Cindersap Forest. This cutscene triggers on or after Spring 11 of year 1, by entering the Forest between 9am and 4pm on a Monday, Thursday, or Sunday.