Perhaps it's a community building. Naruto is a very community - oriented person. He could own a building that serves as a place for the villagers to gather, like a community center. There could be a cafeteria, a place for kids to play, and rooms for meetings and celebrations.
He could transform the building into a ninja academy annex. Naruto loves teaching and passing on his knowledge. In this annex, he could offer advanced ninja training, focusing on things he has learned from his own experiences, like dealing with powerful enemies and using the power of friendship and determination.
One possible storyline could be Naruto renovating the apartment building to make it a better place for the villagers to live. He might use his ninja skills to do some unique construction work.
There could also be external threats. Maybe other villains or rival ninja clans see the apartment building as a target. They might try to take it over or destroy it to undermine Naruto's influence in the village. Naruto would then have to use his combat skills and strategic thinking to protect his property.
In the fanfiction, he might own a building that is actually a magical library. Since Harry has always been interested in knowledge, this building could be full of rare and ancient magic books. It could also be a place where wizards and witches from all over come to study and research, and Harry, as the owner, would be in charge of protecting these precious knowledge resources.
He could turn it into a training ground for young wizards and witches. Teach them how to defend themselves against the dark arts, just like how he was taught at Hogwarts.
I don't think it's a super popular concept in the Naruto fanfiction community. There are so many directions fanfiction can take. Some focus on romance, others on battles or new powers. While Naruto's relation to Konoha is fundamental, the idea of him 'owning' it in fanfiction is not something that seems to be everywhere. That said, there may be some fanfictions out there that play with this idea, but compared to things like exploring Naruto's heritage or his growth as a ninja, it's probably not as prevalent.
Maybe he buys a training building so that he can practice his ninja skills without being disturbed. It could have special facilities for his shadow clone jutsu practice.
Maybe he owns a ninja training academy as a business. Since he is a powerful ninja, he can pass on his skills and knowledge to the next generation. He could offer different levels of training courses for both beginners and more advanced students.