WARNING: The following article contains spoilers for Fruits Basket.
Members of the Sohma Family are cursed to transform into one of the twelve animals of the Chinese Zodiac (plus the cat and "God") whenever they're hugged by a member of the opposite sex. On top of being an abnormal transformation that alienates them already from the rest of society, it also makes relationships incredibly difficult to maintain. While some, like Ayame, tease that they can still have sex without ever hugging, this doesn't make it any less difficult to maintain physical relationships.
This leads to an extra curse those chosen by the Zodiac must endure. Because they cannot be hugged by a member of the opposite sex, the only heterosexual relationships they can form are within the circle of fourteen people inflicted by the Sohma Curse. That means they need to form sexual relationships with other family members. The majority of members of the Zodiac Family have, on some level, committed incest.
Canon Incest Relationships
In Fruits Basket, there are multiple sexual relationships between the tightly knit members of the Sohma family. Kagura is infatuated with Kyo, to the point where she will run halfway across the continent just to be with him. Hatsuharu calls Yuki his "first love," while he also maintains an on-again, off-again relationship with Rin Sohma. Kisa and Hiro Sohma have a puppy-love relationship with one another. Shigure often jokes about a romantic relationship with Ayame while actually sleeping with Akito.
Even some of the relationships that aren't between Zodiac members are forms of incest. Most noticeably, Hatori becomes romantically involved with and almost marries his distant cousin Kana. The Sohma compound is massive and contains many houses filled with people, and the Zodiac members hardly socializes outside of the family.
That is not to say that all the relationships end up being within the family or that they need to do so. Ayame ends up forming a romantic bond with his coworker Mine, later having two children with her in the sequel Fruits Basket Another. Yuki, though he first is romantically interested in Tohru, ends up becoming partners with Machi, the school council treasurer. And, of course, Kyo ends up marrying and having children with Tohru.
Incest is Relative
Longtime fans of shojo manga, however, might not pay attention to this too much. Relationships between cousins aren't as taboo in Japan, and on top of that, the Sohma family functions more as a clan than a genetically close family. The Japanese word the Sohmas use to refer to one another, itoko, is a catch-all term for "close relative," or a relative that isn't your parent, sibling or grandparent. It is translated as "cousin" in English, which gives it a different connotation. This makes it difficult if not impossible to determine how closely related each Sohma cousin is to one another.
Indeed, the Sohma Family is so massive and sprawling that most of them really don't even know one another. They live in their own compound, which is a living arrangement that was made illegal in post-WWII Japan. This is acknowledged in-canon when Momiji acknowledges their family is not "in today's customs." This positions the Sohma family more like a traditional Japanese clan rather than a family in the modern sense. This also explains how divided the Sohmas are from one another, like how Kana Sohma didn't know Hatori or about the Sohma Curse before becoming his assistant. With such a sprawling family, most people don't express concern over it because, genetically, they're very different.
Even if the level of incest is question, the tragedy is that those with the Sohma Curse are unable to get to know anyone else. Because of the tight grip Akito has on all of them, they're stuck with the same group of people all day every day. Kureno Sohma hadn't even been to a convenience store before the age of 26. Growing up in that environment, stuck on a compound with little interaction outside school or family, of course they would form romantic bonds to their family. Who else are they going to form relationships with?