Death did not represent separation from God until Adam and Eve gained knowledge of evil - until that point, people could live uninterrupted under God's sustenance. The expulsion from Eden represents that sting of death, and that happened on the same day. They were exiled into a life slavery under death, just like the Israelites thousands of years later. "Death" is nothing but separation from life. And on that die they became separated from God, the giver of life.
Compare Exodus 10:
28 Then Pharaoh said to him, Get away from me! Take heed to yourself and see my face no more! For in the day you see my face you shall die!
It emphasizes the certainty of death. With the same certainty Ezekiel says: "The soul who sins shall die" (18:4).
And 1 Kings 2:
37 The day you leave and cross the Kidron Valley, you can be sure you will die; your blood will be on your own head."
41 When Solomon was told that Shimei had gone from Jerusalem to Gath and had returned, 42 the king summoned Shimei and said to him, "Did I not make you swear by the LORD and warn you, 'On the day you leave to go anywhere else, you can be sure you will die'? At that time you said to me, 'What you say is good. I will obey.' 43 Why then did you not keep your oath to the LORD and obey the command I gave you?"
Shimei did not die the moment he left Jerusalem, and Adam and Eve did not die the moment they disobeyed God's warning. But at that moment they became guilty: they earned death. God said they would surely die; Satan said: "You will not surely die".
And we know that Adam and Eve surely died, as we all will surely die. The guilt comes into effect the moment of transgression, but punishment is not effected until culpability has been determined. The same is true for Judgment Day.
I will combine this with JD's question, since the misconception is the same. That we will die or won't die does not refer to this life, but the next. Only someone who does not have eternal life can truly be said to be dead, and only someone who has life with God can truly be said not to have perished. As for remaining on "this earth", God promises to establish a new heaven and a new earth, and His kingdom among us.
Jesus came to provide the sustenance that Adam and Eve forfeited - to be our life with God. Instead being condemned to eternal separation from God, Jesus restored our relationship with God. Jesus carried our guilt, baptized it in death, and so God's mercy rests on everybody who shares in his death by receiving God's Spirit.
John 6
49Your forefathers ate the manna in the desert, yet they died. 50But here is the bread that comes down from heaven, which a man may eat and not die. 51I am the living bread that came down from heaven. If anyone eats of this bread, he will live forever. This bread is my flesh, which I will give for the life of the world.
