How, According to the Bible, Could Christ be God and Man?

Answer: The Word “became” flesh.

Both Catholics and Protestants have historically taught that Christ had a corruptible earthly body, which was sinless by virtue of having no earthly father. Within this body God dwelled. When Christ was resurrected his earthly body was transformed into a spiritual body. Others believe that Jesus and his apostles taught that God was manifested as incorruptible flesh from his conception – literally the Word made flesh as opposed to the Word (God) dwelling in corruptible earthly flesh.¹ This latter view was held by Anabaptists, who were instrumental in establishing the separation of church and state early in US history. Abroad, Queen Mary I earned the title “Bloody Mary” by burning Anabaptists at the stake by the thousands. Religious leaders tried forcing them to confess with heart and mouth, that Christ had assumed his flesh and blood from the substance of the flesh and blood of Mary.² King James' translators even went so far as to render early versions of the Bible, which had correctly translated Galatians 4:4 (God sent forth his Son, born of a woman) to read God sent forth his Son, made of a woman.

The Apostle John taught that Christ, which they heard, and saw with their eyes, and handled with their hands, was the same as that which existed from the beginning as the Word of Life.³ Similarly, Jesus taught the Jews, saying: You are from beneath; I am from above: you are of this world; I am not of this world. I said therefore unto you, that you shall die in your sins: for if you believe not that I am he, you shall die in your sins.⁴ Paul made this same distinction: The first man (Adam) is of the earth, earthy: the second man (Christ) is the Lord from heaven .⁵ According to the John, he and his followers were able to discern “Antichrist” spirits that confessed not that Jesus Christ is come in the flesh .⁶

Anabaptists held that Christ having flesh like ours, except from heaven – the true bread which came down from heaven (John 6: 31-35), was essential to a consistent view of Christ.⁷ Adam's body returned to the earth because it was made from the earth; Christ's body did not because it came not from the earth. He was born of a virgin because his body was not of the earth. Christ's body is literally the Word of God; therefore, we must eat of his flesh and drink of his blood to have eternal life.⁸ Those that saw him literally saw the Father.⁹ Resurrected bodies of saints come from the seed of the Word of God , which is sown in the earth (earthly bodies) and raised to be like Christ.¹⁰   

References
¹Jn 1:1-14
² van Braght, T.J. The Bloody Theater or Martyrs Mirror of the Defenseless Christians, 5th ed., 1950. Scottsdale , PA ; Waterloo , Ontario : Herald Press. p.1019.
³ 1Jn 1:1-2
⁴ Jn 8:23-24; See also Jn 3:13
⁵ 1Cor 15:45-50; Compare Col 2:9
⁶ 1Jn 4:1-3; 2Jn 7; Compare Mk 16:17-18; Jn 8:23-24
⁷ See, e.g., 1Cor 15:35-50
⁸ Jn 6:53-58
⁹ Jn 14:9
¹⁰ Mt 13:37-38 ; Mk 4:14; 1Jn 3:1-2