Is Hell eternal or do those who do not choose Christ just cease to exist?

Question: How should we think of hell from God’s perspective?

Answer: There are four words used in the Bible for the place of the dead, Sheol, the Hebrew term for the grave and also for the place of departed spirits, Hades, the Greek version of Sheol, the Abyss (used for the place of the dead in Romans 10:7, but usually reserved for a place of judgment for demons, Luke 8:31; Rev. 9:1,2, 20:1) and Gehenna, the term taken from the Valley of Hinnom just south of Jerusalem where trash was perpetually burned. This last word refers specifically to the final place of torment, most properly translated “hell” (Matthew 5:22,29,30; 10:28; 18:9, etc.)

Matthew 25 is instructive in that it tells us that the eternal fire (the word Gehenna is not used here) was prepared by God originally for Satan and his angels (Matthew 25:41-46), but that unbelievers will end up there as well. In verse 46 it says they will “go away to eternal punishment, but the righteous to eternal life”. The same word “eternal” describes the character of the punishment of the wicked as well as the life of the righteous. This could argue that it means the punishment of hell is eternal, and that the soul will exist in this place of punishment forever.

In Mark 9:43 Jesus describes hell as the place where “the fire never goes out” (He calls it “eternal fire” in Matthew 18:8). There would be no need for the fire to be perpetual if at some point there was no one there to suffer its flames. By the way, it is supposed that the flames are not literal flames that require oxygen and fuel (material) to burn, since the souls of the unbelieving dead are not material (Luke 16:22-24 depicts the rich man in torment in Hades). Even their resurrected bodies will be theirs forever in the Lake of Fire (Revelation 20:11-15) and not be fuel to burn up. This suggests that the fire is more psychological in nature, but nonetheless punishing. Jude mentions the punishment of eternal fire” (v.7). Paul says unbelievers “will be punished with everlasting destruction and shut out from the presence of the Lord and from the majesty of his power” (2 Thessalonians 1:9). How can the punishment be everlasting destruction if the destruction is completed?

Of course, it is hard for us to fathom the justice of this at this point in our lives. We don’t see as clearly as God sees yet and do not appreciate His just retribution, because we are often unable to fathom the heinousness of our own rebellion. One day we will feel the justice of God’s pronouncements of judgment.

For further reading: https://thimblefulloftheology.com/where-no-man-has-gone-before-alternative-views-of-the-souls-destiny/