Having ranted about how people use "Both/And" too often and inappropriately, I am now going to play the "Both/And" card myself. ;)
In regards to the question, "What happens to people in Hell?" the traditional answer is that they are in eternal conscious suffering and the non-traditional answer (or at least the one that I care about for this discussion) is that they are annihilated (possibly after suffering some finite period of suffering).
I would like to raise the possibility that the answer is "Both/And". I should here note that I do not (yet) believe that that is the actual answer. Rather, I want to show that "Both/And" is a viable option as (contrary to all appearances) the two positions are not mutually exclusive.
Consider black holes. (Note: I am not here saying that Hell is a black hole. That black holes are hot, dark and unescapable is merely coincidental. I'm just trying to give an example where a person could be in eternal suffering and also be annihilated after a finite time.) Black holes warp space/time in their vicinity to the point where, at the event horizon, they've essentially torn it. This leads to some rather strange results.
(Warning, the following discussion assumes some knowledge of the Relativity. I'll try to keep it simple, but my apologies in advance if I fail to keep it at a low enough level.)
From the frame of reference of a person falling into a black hole, the experience would be painful but finite in duration. That person (we'll call him "Bob") would be either crushed by tidal forces or fried by radiation. Let us say for the purposed of this example that Bob dies at the moment he reaches the event horizon, but that he consciously suffers until he dies. No one really knows what happens after you cross the event horizon of a black hole, but annihilation is a good a guess as any. (It is, in fact, a better guess than most.) From Bob's frame of reference, he suffers for a finite period of time, then dies, and then is annihilated.
Now let us look at a different frame of reference...
Alice is watching Bob fall into the back hole from her spaceship a safe distance away. From her frame of reference, Bob never actually reaches the event horizon. Due to time dilation (i.e. the warping of space/time around the black hole), Bob, from Alice's frame of reference, falls more and more slowly towards the event horizon. He gets ever closer, moving ever slower, but he will never actually fall into the black hole. From Alice's frame of reference, Bob endures eternal conscious suffering.
Keep in mind that neither of these is "what actually happens" while the other one is "just an illusion". They are both "what actually happens" according to relativity theory. From Alice's frame of reference, Bob really does endure eternal conscious suffering, and from Bob's frame of reference, he really does endure a finite period of suffering followed by annihilation. Neither one is "more right" than the other; each version of the events is correct in regards to the appropriate frame of reference.
Thus it is possible for both the traditional view and the annihilationist view to be correct. That the Bible would more often use the language of eternal conscious suffering makes sense as the authors are presumably not writing from the frame of reference of descending into Hell. But such language does not necessarily exclude Annihilationism from the frame of reference of the person actually descending into Hell.