When and where is "Hamlet" set?
Shakespeare doesn't actually specify when "Hamlet" is supposed to take place. It's obviously set in Denmark, and, despite a recent trend of claiming that it's definitely set in the 12th century, Shakespeare isn't specific.
We know that Shakespeare's company seemed to perform the play in modern dress with a few trappings of costume: so, for the Peachum drawing we have of a performance of the Roman-set "Titus Andronicus", the actors are dressed in Elizabethan doublet and house costume, with a makeshift sort of toga and crown set on top.
Most likely then, is that Hamlet (particularly as the play itself seems to deal with the conflict of Protestantism and Catholicism) was played as a modern-dress play, but also one set in the Denmark of the past.
There are Elizabethanisms here and there in the text - not least the company of players, whose business suffers from the onset of boy players, which was simultaneously occuring in the Elizabethan theatre as Shakespeare wrote the play. One version of Hamlet has Ophelia enter, mad, with a lute: an almost symbolically Elizabethan instrument.
Hope it helps!