It is often said and believed by many that our souls are immortal - that God gave us a spirit/soul that cannot die when he created us, and that death was not even a concept or reality before Adam and Eve sinned in the Garden of Eden. Despite this idea being a fairly "recent" concept (in terms of history and theology), and it stemming from Greek philosophy, it's also not supported by the Biblical text - especially in Genesis.
Lets look at the creation account in Gen 2:7—
then the Lord God formed man from the dust of the ground, and breathed into his nostrils the breath of life; and the man became a living being.
The word "being" in that verse, or "soul" in older translation, comes from the Hebrew word "nephesh". This is defined in Strong's lexicon as: "soul, self, life, creature, person, appetite, mind, living being, desire, emotion, passion" - but never as inherently immortal. We should also note that verse seven in Genesis 2 also states that man became a living being/soul once God had breathed life into the newly created body; as far as the Hebrew language used in Genesis allows, it only shows that God created people as mortal beings, and that a "soul" is nothing more than a living, breathing creature (I say "creature" instead of "human being", as the same nephesh word is used of animals too in the Creation account - but that's another topic!).
To the Hebrews, the body and soul were one 'unit' that worked together, rather than two separate entities; there was "no idea of an immortal soul living a full and vital life beyond death" and many Psalms and older Biblical texts reflect this idea too (cf. Psalm 6:5; Psalm 88:3-12; Psalm 146:4; Psalm 115:17; Job 14:10-12; Job 3:11-19).
With that in mind, the threat of death by God makes more sense since Adam and Eve would actually be able to comprehend such a concept if it was already a reality. They obviously appear to have been created with unlearned knowledge since they can speak and talk about things (either that, or Gen 2 skips a bunch of narrative of them growing up and learning everything), so if physical death didn't exist until they sinned, what good would it be to say "you will die" in Genesis 2:16-17?
The trick of the serpent was to convince Eve that she wouldn't die by eating from the Tree of Knowledge - except that they did indeed die because they could no longer attain immortality; they stayed mortal as punishment. After they ate the forbidden fruit, God 'evicted' them from the Garden in case they "might reach out [their] hand and take also from the tree of life, and eat, and live forever" (Gen 3:22) - but if they were already immortal then this would surely be of no concern to God.
Their sin separated them from God as they would now not live forever with him (Ez 18:4; Ez 18:20). The concept of a spiritual aspect to people that is separate from the physical body seems to come in later, or is mixed into the concept of Sheol. Was Sheol was simply a euphemism, or caricature, for the grave and death which developed over time; or did the ancient Hebrews think of a literal life-after-death scenario (as Luke 20:38 would suggest in retrospect)? Either way, from reading through the Old Testament, you begin to see it as "a place of darkness to which all the dead go, both the righteous and the unrighteous, regardless of the moral choices made in life, a place of stillness and darkness cut off from God" (Mercer Dictionary of the Bible, p. 819).
Only in Dan 12:2 do we find any idea of people coming back from the place of the dead:
Many of those who sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt.
The concept of the Resurrection is a relatively late concept, as Daniel is the latest book of the Hebrew Bible, and as we get to the 1st Century onwards, all were agreed that there "was no resurrection" - except for the Jews, of course.
What we can see in the Resurrection is that God wants us to be with him in a physical sense, not in some disembodied state. This is why we were created; and why the Jewish concept of blessing and reward from God is a long physical life (eg. Exodus 20:12); and also why the New Testament is concerned with the restoration of all creation (cf. Col 1:15-22; Rom 8:19-23) - and not an escape from it, through the resurrection!
1 Tim 6:16 tells us that it is God "alone who has immortality", so clearly we mere mortals can't also be immortal, can we? Throughout the New Testament letters and the in the Gospels, it is made clear that eternal life is a gift which we are given through faith in Christ, and something we attain - not something we are born or created with (see: John 5:24; John 3:14-16; Romans 2:6-7; Romans 6:22-23; Titus 3:7; 1 Jn 2:25; Jude 1:21). In Revelation 2:7, Jesus speaks of the Tree of Life as a reward and as a way in which eternal life is given as he promised:
To everyone who conquers, I will give permission to eat from the tree of life that is in the paradise of God.
Is this how we attain our immortality? Jesus came to restore creation back to what was originally intended, and if eternal life was the plan (as it appears to be in Genesis) then it makes sense that, through Christ, we are now given access to the Tree.
It's also interesting to note that there was no forbidding of eating from the tree of life until after Adam and Eve ate from the Tree of Knowledge. The curse on the woman was to increase her pain too, not create pain (another misconception about the Fall - a life without pain seems to be more dangerous than with it!).
In light of this, it seems to me like death and pain did exist from the start as a part of the original creation, but that maybe we could withstand pain better or that God was going to allow immortality to all in some way too, since it would appear our lifespan's were much greater originally (see Gen 5:3 for example). Spiritual death would also make more sense in the context of this, as our newfound wisdom and knowledge which made humans "become like" God (Gen 3:22) and that then produces pride, which is shown throughout Scripture to be something that sets us against God, and puts us into disharmony with our Creator - this is what happens at Babel (Genesis 11:1-10).
Looking across all of Scripture, it does seem like the idea of Sheol being more than just a physical grave develops and becomes an actual spiritual 'abode' for a spiritual aspect of humanity. This is often called "Progressive Revelation", and is "the concept that the sections of the Bible that were written later contain a fuller revelation of God compared to the earlier sections", which in this case would assume that we've always had a spiritual state, but that earlier authors just didn't know about it and assumed death was the end.
With that said though, Jesus's warning in Matthew 10:28 where he states that God can not only kill the body, but also the soul, ought to give us pause for thought. If the soul survives death, it doesn't mean it is immortal since God can still destroy it!
In Jesus we see "the reflection of God’s glory and the exact imprint of God’s very being" (Heb 1:3), and so the New Testament and the teachings of Jesus ought to be the way in which we view the Old Testament's views on death and afterlife - which would lead us to viewing ourselves as having a "soul" in the spiritual sense, and that the death in Adam was also a spiritual death that only Christ can reverse by sealing us and renewing us with Holy Spirit in this life (2 Cor 5:17; Eph 1:13; Eph 4:30), and then by looking forward to the hope of the resurrection to come (Rom 6:11; 1 Cor 15:20-22; 1 Pet 3:18).
Jesus did more than restore us spiritually by defeating spiritual death, but also by defeating physical death so that we CAN eat of the Tree of Life and live forever! He beat death COMPLETELY - physically and spiritually (Isaiah 25:8; 2 Tim 1:10; 1 Cor 15:26; Revelation 20:14; Revelation 21:4); more so than we ever imagined, and now God grants us life everlasting so that we may dwell with him, and he with us, as originally intended (2 Cor 6:16; Rev 21:3)!