You know, the whole can't get sick or injured thing?
She mentioned it before when she first gave him a massage and he wanted to give one back already. That was over 20 chapters ago.
People seem to have short memories about this shit.
And she isn't the moon or lunarian or from space. She has told Chitose, someone that should know the truth of her, straight to her face that she is a normal human. There's other things, like the fact that the Q&A sessions at the end of volumes had one instance where she said her hometown was the capital of antiquity (clearly referring to an ancient capital of Japan, such as Nara City or Asuka), too, but still... The only thing that's special is that she probably received the immortality elixir from the Tale of the Bamboo Cutter, and that's the connection to Kaguya-hime. Nothing more. It's not as fantastical as you all made it out to be, I guarantee it and I look forward to reaching the chapter where I'm proven right on this.
Just about all the evidence from the past 9 volumes has pointed to her being a native of the Nara Prefecture from over a thousand years ago when the events of the Tale of the Bamboo Cutter took place, and instead of the elixir being burned atop Mt. Fuji, it somehow was given to Tsukasa instead. Likely her father was one of the people tasked with its destruction, and he defied imperial order to give it to his daughter. I'm quite confident this is essentially what happened.