What does Hamlet mean when he calls Denmark a prison?

Answer 1

Polonius leaves, and Rosencrantz and Guildenstern (who have been watching) enter. Hamlet realizes right away that they have been sent for. They share a dirty joke about "Lady Luck's private parts" that would have been very funny to Shakespeare's contemporaries, and Hamlet calls Denmark a prison. When they disagree ("Humor a madman"), Hamlet says "There is nothing either good or bad but thinking makes it so. To me it is a prison." (Note that Hamlet is obviously not referring to the idea that there are no moral absolutes -- as do certain contemporary "multiculturalists".) The idea that attitude is everything was already familiar from Montaigne, and from common sense.
The spies suggest Hamlet is simply too ambitious. This is ironic, since they are the ones who are spying on their friend for a king's money. Hamlet replies, "O God, I could be bounded in a nutshell and count myself a king of infinite space, were it not that I have bad dreams." The friends continue to play on the idea that Hamlet's ambitious are being thwarted, sharing some contemporary platitudes about the vanity of earthly ambitions. But it seems (from what will follow) that Hamlet's remembering the time when the world seemed like a much happier place. Hamlet then questions the men again about the purpose of their visit. If they actually cared, they would say, "Your family asked us to come. We are all very worried about you." Instead, they pretend they just dropped by, which is stupid. Only when Hamlet asks them "by the rites of our fellowship" (i.e., by our secret fraternity ritual) do they have to tell the truth. (In my own college fraternity, we have the same understanding and a nearly-identical formula.)
Hamlet levels with his friends. There was a time when the beauty of the earth, the sky, and the thoughts and accomplishments of the human race filled him with happiness. (All of this is good Renaissance thought.) Now he has lost his ability to derive enjoyment, though he knows the earth, sky, and people should still seem wonderful. They seem instead to be "the quintessence of dust". Anyone who's experienced depression knows the feeling. "Quintessence" ("fifth essence"; compare Bruce Willis's "Fifth Element") was an idea from prescientific thought -- a mystical substance that made fire, air, water, and earth work together, and supposedly what the planets and stars were made of.

Answer 2

He means that Denmark is really bad and he thinks it is a realy bad place. A prison is a prison is a really horrable place so he was comparing the two. He is saying that thay are alike because thay are both bad places.

Answer 3

OH COME ON. HAVE YOU EVER BEEN TO DENMARK. 'NUFF SAID.