What is the ruling on a young man staying in the home of his maternal aunt who lives with her husband and adult son?

There is no legal impediment, from a Sharia (definition) perspective, to visiting relatives such as the maternal aunt's home, provided that one adheres to the Islamic etiquettes and rulings. It should be known that the maternal aunt's daughters are non-mahram (definition) women (strangers to whom marriage is permissible), and it is forbidden to be alone with them or to gaze at them, based on the Almighty's saying: "Tell the believing men to reduce [some] of their vision and guard their private parts," and the Prophet, peace and blessings be upon him, said: "No man is to be in seclusion with a woman unless there is a mahram with her." If the maternal aunt's daughters are mahram through suckling, then they are like sisters. As for the maternal aunt herself, she is always a mahram, based on the Prophet, peace and blessings be upon him, saying: "The maternal aunt is like a mother."