From Genesis 2 (KJV):
8 And the Lord God planted a garden eastward in Eden; and there he put the man whom he had formed.
9 And out of the ground made the Lord God to grow every tree that is pleasant to the sight, and good for food; the tree of life also in the midst of the garden, and the tree of knowledge of good and evil….
15 And the Lord God took the man, and put him into the garden of Eden to dress it and to keep it.
16 And the Lord God commanded the man, saying, Of every tree of the garden thou mayest freely eat:
17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
In Genesis 3:
2 And the woman said unto the serpent, We may eat of the fruit of the trees of the garden:
3 But of the fruit of the tree which is in the midst of the garden, God hath said, Ye shall not eat of it, neither shall ye touch it, lest ye die.
4 And the serpent said unto the woman, Ye shall not surely die:
5 For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil.
I presume you all know what happens next so, skipping ahead:
22 And the Lord God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever:
23 Therefore the Lord God sent him forth from the garden of Eden, to till the ground from whence he was taken.
24 So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life.
Reading this text on its own, unencumbered by 2000 years of Christian theology, it is very clear what happens in this story. Adam and Eve were not immortal at any point. The threat that they would die was a lie told to dissuade them from eating of the Tree of Knowledge, which God did not want them to do because they would then become more like him, which he seems to have found threatening.
He expelled them because there was another tree in the garden, the Tree of Life, that would also make anyone who ate its fruit immortal, and God worried it would only be a matter of time before Adam and Eve ate from that, too, and further challenged his special status. So he kicked them out of the garden, and installed guards to prevent anyone from approaching the Tree of Life as an added precaution.
I don’t really see any other reasonable interpretation of the bare facts of the story.