The BFG takes place primarily in England and Giant Country, in modern times.
In England, the story begins in the orphanage where Sophie lives, and she and the BFG later travel to London to visit the Queen. By the end of the story, they both live in Windsor Park.
There are anecdotal accounts of the man-eating giants traveling all over the world as well, such as to Sweden and Turkey, although Sophie and the BFG do not actually go to these places.
After the BFG abducts Sophie, much of the action in Giant Country simply takes place in his cave in the side of a moutain. He also takes Sophie to Dream Country, although this is more of a quick side-trip that elaborates on the BFG's job of spreading dreams.
The story takes place in modern times, although a specific year is never mentioned. However we can deduce this through the appearance of technologies like the telephone and helicopter, and the giant's mention of airplanes and bombs.
The BFG takes place in modern-day England, as well as in Giant Country. In Giant Country, the Giant lives in a cave within a mountainside, whose entrance is protected by a large round stone. Giant Country is described as a "great yellow wasteland with its blue rocks and dead trees" (page 33). There, mean, ugly giants (all of whom are male) live beneath the glare of the blazing sun. There, only what the Giant calls "one extremely icky-poo vegetable" called the snoozcumber grows (page 48). Part of the book also takes place in Dream Country, where the Giant takes the girl he has captured, Sophie. In Dream Country, the Giant captures dreams with a butterfly net and places them in jars.
The book takes place in recent times, as Sophie and the Giant visit the Queen of England in Buckingham Palace. Elizabeth is the only recent queen of England, and she has been on the throne since 1952. In addition, the queen sends helicopters (a relatively modern form of transportation) to Giant Country to capture the mean giants, so the book takes place in modern times.