The Apollo missions brought back several kilos of moon rock from the moon. Some of it was analysed to test what the moon is made of, some of it was given as a diplomatic gift to different world governments. If you can motivate it with a good reason (i.e. scientific research) you could probably ask NASA if they can provide some for analysis.
No one owns the moon itself, and I think it is also protected under international treaty that says no one can claim ownership.
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