I honestly do not believe so. The idea that this is some form of hell or a punishment implies that there is some larger force that is punishing us, some entity that actually wishes to see us suffer. I know that certain people may believe this to be the case (and I don't want to dismiss them, faith in divine power is a valid perspective for making sense of our bizarre existence), but all the evidence that we actually have available points to a universe that simply doesn't care. We could lead neutral lives, achieve immense happiness, or suffer unbearably for millennia, and to the greater scope of the cosmos, it's all the same.
I can definitely understand why people would want to believe the world exists to torment them, though. There is so much suffering in it. The idea that this is some sort of punishment for us ties all that suffering up with a neat little bow. It provides an easy and digestible answer for why so much suffering exists. This answer doesn't fix it, and it doesn't make it any easier to bear, but it at least makes all the suffering in the world make sense. For some, it's just a little more comfortable to be able to have a why for their suffering, something to explain it, instead of enduring pain and torment for seemingly no real reason at all.
So with that said, let me provide an alternative explanation.
Life...is fragile. The conditions in which life can exist are extremely narrow, the complexity involved in even the most basic organism attempting to remain 'alive' just boggles the mind. Dead things have it easy. They can be anywhere, doing anything, and remain dead. The crushing weight at the center of the earth. The heart of a star. A black hole. Dead things don't care. They can enter, leave, or return to any of these environments and remain just as dead as they are. Dead stuff doesn't need to care.
Life, though? Life has to walk an incredibly narrow tightrope to remain in the thin sliver of conditions and circumstances which allow it to stay alive, with death surrounding it on all sides. Think about the things that make you suffer. The physical pain of an injury, the pangs of hunger, the emotional pain of isolation or judgment, the existential dread of knowing that none of it will ultimately matter in the end. Each of these presents a threat, something that can either push you off that thin sliver we call life (injury, hunger), or make it more difficult to stay balanced on it (social isolation, despair). Pain is part of that balancing act, a force designed to push us away from those things that threaten our precarious perch on this extremely narrow knife's edge of a tightrope. And since those things are everywhere, all the time, suffering is very often found everywhere, all the time.
This also explains why other people can very often be a huge part of our suffering. We're all in this together, perched on this same tightrope. As one person fails and spins their arms to keep their balance, they have a tendency to knock into the people around them, upsetting their balance and bringing them closer to some form of threat, resulting in pain. No one is immune from this; we all cause pain to others around us at some point or another in an attempt to reduce our own. It's an unavoidable part of the human experience.
So no, this isn't a hell. We weren't put here to suffer, no one is punishing us, and we don't deserve our shared pain. It is just a common-sense manifestation of an incredibly fragile system trying to keep itself going, and avoid the things that could stop it. And the only thing any of us can really do, apart from try to escape it ourselves (ctb), is to do what we can to try and reduce the suffering of those around us.