John and Gillian's world

"John and Gillian's world" was the setting of Dr. Who's adventures with his grandchildren John and Gillian. (COMIC: The Land of Happy Endings)

Accounts differed as to whether these adventures took place in reality (PROSE: Beware the Trods!, The Man Who (Nearly) Killed Christmas) and John and Gillian were truly the Doctor's grandchildren (COMIC: The Klepton Parasites, et al.) or simulacra, (POEM: The Five O'Clock Shadow) or if they took place in the Land of Fiction (PROSE: Conundrum) or the Doctor's dreams. (COMIC: The Land of Happy Endings)

History
to be added

Nature
One account suggested that the "Dr. Who" who travelled with his two grandchildren existed in the Land of Fiction. While visiting, the Seventh Doctor encountered a "John" and "Gillian" who claimed he was their grandfather in whose company they had fought the Kleptons and the Trods. They tried to make him remember that his name was Dr. Who, to which he flatly replied with denial. When they pressed further and asked if he remembered them, he said that he didn't. (PROSE: Conundrum)

Another, unrelated account stated that one version of the Eighth Doctor who had recently encountered Spring-heeled Jack (COMIC: The Curious Tale of Spring-Heeled Jack) had dreams in which he had adventures with young versions of John and Gillian. The Doctor believed that he dreamed this world to envision a reality where "villains are naughty, not evil", "people never die", and "promises are never broken".

However, by one reading of this account, the world of John and Gillian potentially was real, as both Eighth Doctors dreamed of each other, meaning the "Spring-heeled Jack" reality could be the dream world. (COMIC: The Land of Happy Endings)

It's important to note that the above two sourced were not consistent with John and Gillian eventually growing older and leaving for Zebadee University. (COMIC: Invasion of the Quarks, Beware the Trods!)

In yet another account, John and Gillian did indeed physically exist and travel with the Doctor, but they were mathematically-computed simulacra given reality by the TARDIS based on the Doctor's wishes. They were only two of "thousands" of companions whom the Doctor had created out of thin air, perfect and unchanging, to travel at his side without fear of death, in an effort to avoid responsibilities and anxiety. These evading techniques led to the creation of a psychic entity known as the Five O'Clock Shadow, which the Doctor ultimately escaped by using the same TARDIS functionality to give life to perfectly angst-free versions of himself and Susan. (POEM: The Five O'Clock Shadow)