No. Sheol in the OT, and Hades in the NT, both refer to the same place – the place of the dead. This is the grave. When people die, the breath of life goes back to God who gave it, and we’re simply asleep until the Second Coming of Jesus (see What happens when we die? for further study on this).
Gehenna in the NT refers to a valley outside of Jerusalem where everyone dumped their trash. It was always on fire. Jesus references this a few times – one example is when he talks about it being better to lose one part of you to that valley than for your whole body to be burned in it. He’s talking about when Satan and his angels, plus all the evil people, and death itself, are thrown into the lake of fire at the very end and using an example that people of that time could relate to.
The eternal punishment in the Bible has been distorted and incorrectly referenced. It is used to push the false doctrine of an actual, literal Hell that exists forever, where people will be eternally tormented. That is not the case at all. God is a God of Love. The eternal punishment mentioned in the Bible is eternal separation from Him. If you choose not to follow and serve Him, ultimately you will be no more. Even the memory of this current Earth and all the wicked, sin, pain, misery, and death will not be remembered.
Just think about this for a moment. Does it make sense for God to eternally punish someone who chose not to repent, who lived for an average of only 70-80 years on this Earth? Does it make sense that you would look down on your loved ones that didn’t make it in the future, to forever look on them in agony in eternal punishment? Does that sound like a loving God to you? Of course not. The wicked have made their choice, are consumed by fire at the end, and are ultimately no more. They simply turn to ash and are gone forever:
And ye shall tread down the wicked; for they shall be ashes under the soles of your feet in the day that I shall do this, saith the LORD of hosts.
Malachi 4:3
The term unquenchable or eternal fire is used to show that everything will be completely consumed, nothing will snuff it out until its work is complete. It’s a figure of speech. We use figures of speech in our daily conversations. “It took forever to get there” for example. It didn’t literally take forever.
Once everything that can be burned is burned up, it doesn’t keep burning. For example, Sodom and Gomorrha are not still burning to this day:
And turning the cities of Sodom and Gomorrha into ashes condemned them with an overthrow, making them an ensample unto those that after should live ungodly;
2 Peter 2:6
Even as Sodom and Gomorrha, and the cities about them in like manner, giving themselves over to fornication, and going after strange flesh, are set forth for an example, suffering the vengeance of eternal fire.
Jude 1:7
Who shall be punished with everlasting destruction from the presence of the Lord, and from the glory of His power;
2 Thessalonians 1:0
The wicked are wiped out, destroyed forever. They are not alive, suffering forever in eternal torment. The eternal punishment is simply no longer existing, forever. It will be as if they never existed.