Answering the question first (tldr;)
I see hell as not a punishment, but a double exercise of mercy. On the one hand, God is not withdrawing the goodness he has given us which is our very existence. It would be a worse punishment for him to remove them from existence than it would be for him to leave them where they are. (though, strictly speaking, it would be easy to argue that in order for Him to do this it would involve a contradiction as He would be calling something "not good" which He defined as "good"). The other mercy is His respect for our free will.
The reason hell is eternal (and their is no repentance) is the very fact of this free will. Once we have chosen to separate ourselves from the divine presence so directly and completely, the corruption and bitterness in our hearts only increases.
Whatever we are at death will increase forever after the final judgement. Those in heaven will forever race toward God, completely unencumbered by the flesh. On the other hand, a soul in hell forever races away. It cannot repent because after a thousand years it is more repugnant than it was than it first in.
Why would someone go to hell?
I know that this is not strictly part of the question, but it does have a very strong effect on my view of hell.
Based on conversations I've had with other Christians of late, I think I may need to mention that there is a delight in sin. There is a revelry in corruption that most of us have either forgotten or have never considered. I do remember a time in my life, however, that I actually enjoyed watching the corruption of others. I can totally imagine that in such a state I would have turned my soul away from the divine.
I have also had times when pride has spoken far louder than humility. I was once challenged to act righteously at such a point and I chose not only to reject righteousness, but I also turned from the friends who so desperately tried to turn me from pridefulness. The tremendous feeling of indignation and self-righteousness caused me to fall, and fall hard (it took a year and a half to resolve that act). I can see that easily leading to destruction too.
Finally, I know what it is like to turn from God in despair and swear against Him, expecting that it would result in my death. And at that moment, I did not care what His opinion was for I called Him dead.
I see judgement this way: if I, who know Christ and was taught to love Him in my youth can stray so incredibly far, can I really believe that there are those who have not made similar judgments? If I were to reach such a purifying state as we will see in the eschaton while in those states, I can very easily believe that I would have rejected God perpetually.
What do I base the doctrine of hell ON?
I think I also should address why I do not believe that hell should rightly be thought of as punitive, despite the many images in scripture as a place of "blows."
I find the images of hell in the Bible actually fit the image of systematic "putrification". After death those who have little will find even the little they have taken away (parable of the talents), and the foolish will have their houses swept away in the flood. Since we know that the only thing a man has after death is the condition of his soul, this, to me, means that those who do not go to heaven lose the last semblance of righteousness that they contained. Further, anything which is placed in perpetual, unquenchable fire will be more and more burned up. Even if the level of pain remains constant, that will mean that it will still increase by virtue (he... virtue) of the fact that it has lasted longer (any pain which lasts longer will be worse than any pain which ends sooner).
Further, the testimony of Saints say that hell is a place deprived of love for God (Little Flower once said that if possible, she would even dwell in hell if only to increase the love for God there. How awesome is that?). If love is absent, only its opposite can exist. Such bitterness (or hate, or anger, whatever you would list as what is left when the last vestige of love for anything but the self is removed), combined with the very pains of hell (which are more tortuous than we can even imagine), can only lead to greater corruption.
I think that the most depressing thing about hell (and perhaps the most damning) is that people in hell will likely refuse to even admit to fault. They will blame God because they will not be able to admit to their own faults (for without love, pride grows faster than thistles and thorns). They will not turn from sin, in part, because their very existence is one denial and excuse after another. They will not be able to admit that they have committed a wrong and I would even guess that they get to the point where they would rather spend eternity in hell than admit to the slightest of errors.
Why would removal from existence fail divine mercy?
Two reasons. First God has made us ontologically good. We are His likeness and image. There is no point where God ever removes that reality, and, if He did, it would involve the statement that a person has the ability to change His worth through his thoughts and actions.
This is something which is repugnant to Christian doctrine. If a man has the ability to change his value, then all value is subjective we can start doing things like killing the infirm, ignoring those in need, and catering to the wealthy. No, we have an objective, intrinsic worth, even if we are corrupt. We are worth so much, we are so important to God (even the least of us) that He came in the likeness of sinful flesh to redeem us from our corruption. His blood was not shed that whoever follows Him might have eternal life. He did not die for the righteous, but the sinner. That speaks not to subjective but to objective worth (for you would not sacrifice all for something which had no value whatsoever).
The second reason, the respect for free will, also cannot allow for the destruction of the soul. A soul which is so corrupt likely feels that it has not done nothing wrong. If it has done nothing wrong, then it would absolutely choose existence over non-existence (and I would even argue that no person would ever truly want to stop existing). The corrupt soul will choose hell because it views it as the best option, as perverse as that may sound.