What does it mean that "the two natures of Christ cannot be separated"?

I generally write from a reformed perspective, but I don't think there's anything in this post that other Christians (Oriental Orthodox and Church of the East aside) would disagree with.

The doctrine was first formulated clearly by the Council of Chalcedon:

One and the same Son, the Self-same Perfect in Godhead, the Self-same Perfect in Manhood; truly God and truly Man; the Self-same of a rational soul and body; co-essential with the Father according to the Godhead, the Self-same co-essential with us according to the Manhood; like us in all things, sin apart; before the ages begotten of the Father as to the Godhead, but in the last days, the Self-same, for us and for our salvation (born) of Mary the Virgin Theotokos as to the Manhood; One and the Same Christ, Son, Lord, Only-begotten; acknowledged in Two Natures unconfusedly, unchangeably, indivisibly, inseparably; the difference of the Natures being in no way removed because of the Union, but rather the properties of each Nature being preserved, and (both) concurring into One Person and One Hypostasis; not as though He were parted or divided into Two Persons, but One and the Self-same Son and Only-begotten God, Word, Lord, Jesus Christ; even as from the beginning the prophets have taught concerning Him, and as the Lord Jesus Christ Himself hath taught us, and as the Symbol of the Fathers hath handed down to us.

The church fathers at the council were influenced by Cyril of Alexandria's formulation, which stressed that Christ was "one" "out of two natures."

Firstly, according to Chalcedon, he is of two natures "unconfusedly, unchangeably." That is, his human nature and his God nature did not fold into one another to become a third thing neither human nor God. Instead, he is uniquely both God and man, fully and eternally. He is omnipotent, omniscient, and so on because he is God. He has a body, emotions, and so on because he is human.

But secondly, and more to the point of your question, he is of two natures "indivisibly, inseparably." That is, he is not two people. Jesus, God the Son, is the second person of the trinity, is our redeemer who was born to the virgin Mary and walked among us and sacrificed himself for us.

As far as how his natures are united in his person, that's a difficult question which we will never fully grasp this side of eternity. But Daniel Wallace has some thoughts:

We need to think of the divine attributes in two categories: moral attributes and amoral attributes. The moral attributes are those attributes that speak of God’s morality—justice, mercy, love, goodness, kindness, etc. The amoral attributes are those that speak of God’s sovereignty—omniscience, omnipresence, omnipotence, infinity, eternity, immutability, etc. What is interesting to observe in the Gospels is that a clear line of demarcation can be seen with reference to Jesus: he never fails to function on the level of the moral attributes, but frequently does not display the amoral attributes. In other words, the moral attributes seem to be “hard-wired” to his human consciousness, while the amoral attributes seem to be subject to the guidance of the Holy Spirit and come to the human conscious level at the Spirit’s choosing. At the same time, since he does occasionally demonstrate the amoral attributes, there is no denying his deity. Although Jesus Christ has both a human and divine nature, he is not two persons. He has one consciousness. It is not enough to say that his divine nature does not always operate at the level of his human consciousness. Why? Because it is only the amoral attributes that fit this description.

What would it mean for his natures to be separable? You'd have Nestorianism. Nestorian thought went thus:

Nestorius tried to answer a question considered unsolved: "How can Jesus Christ, being part man, not be partially a sinner as well, since man is by definition a sinner since the Fall". To solve that he taught that Mary, the mother of Jesus gave birth to the incarnate Christ, not the divine Logos who existed before Mary and indeed before time itself. The Logos occupied the part of the human soul (the part of man that was stained by the Fall). But wouldn't the absence of a human soul make Jesus less human? Nestorius rejected this proposition, answering that, because the human soul was based on the archetype of the Logos, only to become polluted by the Fall, Jesus was "more" human for having the Logos and not "less". Consequently, Nestorius argued that the Virgin Mary should be called Christotokos, Greek for "Birth Giver of Christ", and not Theotokos, Greek for "Birth Giver of God".

Nestorius believed that no union between the human and divine was possible. If such a union of human and divine occurred, Nestorius believed that Christ could not truly be con-substantial with God and con-substantial with us because he would grow, mature, suffer and die (which Nestorius argued God cannot do) and also would possess the power of God that would separate him from being equal to humans.

That is not orthodox, and was condemned at the Council of Ephesus (prior to Chalcedon).

The inseparability of Christ's two natures also makes one other thing clear. Contrary to what one may hear in some corners of Protestantism, Christ's natures did not separate on Calvary. Whatever happened as he cried out from the cross, he did not cease to be God. If God could cease to be God, our whole theology as Christians is in utter ruin. If Christ was not God to begin with, our whole salvation was a sham. We must affirm that Christ was both God and man and that those natures are united and inseparable.

Martin Luther and Charles Spurgeon are both quoted as saying things to the effect of, "God forsaken by God -- who can understand that?" I certainly don't. But read the full Psalm 22 which Jesus was quoting and you'll see that, while he (the Psalmist, in anticipation of Jesus) begins with a lament at God's forsaking him, right around verse 19 he pivots to a message of hope of deliverance. Other commentators have pointed out that, though Jesus doesn't say "Father" as he does in other places, he still is addressing him as "My God." Significant because it acknowledges a personal relationship. I don't wish to belittle the enormity of suffering, helplessness, and desolation that Jesus experienced on the cross (God forbid I do such a thing!) but I do want to make sure two things about it are clear, even if other things unfortunately remain unclear: Jesus did not cease to be God, but Jesus was crying out to God the Father for having forsaken him.