As far as place is concerned, the story progresses through a variety of locations around and on what is known by sailors as 'Ship Trap Island' in the Caribbean Sea - a place feared by them. The scene is first set on a yacht as it sails through the waters of the Caribbean Sea, as is made clear clear in the following extracts:

There was no sound in the night as Rainsford sat there but the muffled throb of the engine that drove the yacht

"...even you can't see four miles or so through a moonless Caribbean night."

Next, Rainsford finds himself in the water after he loses his balance and falls overboard - 

....blood-warm waters of the Caribbean Sea dosed over his head.

He then finds himself on the shore of Ship Trap Island an area which Whitney had called "a God-forsaken place."

The island is much feared and causes anxiety amongst the sailors since it has a bad reputation. Rainsford discovers that the island probably deserves its pseudonym when he reaches it and sees the jagged crags jutting out from its side and its thick jungle.

After much struggle, Rainsford reaches an enormous chateau, the dwelling of General Zaroff. The chateau has a 'medieval magnificence ' about it and clearly depicts that the general is wealthy. The general calls the island and the chateau 'his preserve.' It is here that Rainsford is entertained, intrigued and later shocked by the general's tales about hunting, especially when he tells him about hunting humans - 'the most dangerous game'.  

Rainsford soon becomes a victim of General Zaroff's lust for hunting human prey and finds himself in the jungle. He eventually manages to outwit the general and, in the end, is back in the chateau where he confronts General Zaroff on his own terms.

The time frame is clearly some period after the Russian Revolution of 1917. General Zaroff makes this clear in his reference:

"After the debacle in Russia I left the country, for it was imprudent for an officer of the Czar to stay there. Many noble Russians lost everything. I, luckily, had invested heavily in American securities, so I shall never have to open a tearoom in Monte Carlo or drive a taxi in Paris."

This is further supported by the fact that he says,

"Ivan once had the honor of serving as official knouter to the Great White Czar..."

The time period of "The Most Dangerous Game" is the 20th century when game hunting was a popular past-time for the wealthy class.  The story involves three locations: the yacht Rainsford is sailing on, Zaroff's chateau, and the jungle lands on ship-trap island.  The island is located in the Caribbean Sea towards Rio.