In what sense is God one; do the three persons share one body?

God does not have a body. His being is pure spirit. The Father, Son, and Holy Spirit are three distinct persons sharing one divine substance. They are one in nature, not in body. The human analogy of people sharing human nature gives only a faint idea of the mystery. For detailed understanding, read the Council of Toledo. It defines the relations and unity of the Trinity better than any later text.