According to Nasu, Fate/EXTRA is a JRPG which takes place set in an alternate universe that split from the Fate/stay night timeline in the 1980s to 1990s.

The Story is set in the in a school called Tsukumihara Academy, a fictional place existing with an artificial environment called the Moon Cell. There, the Moon Cell orchestrates a replication of the Holy Grail War using a system called Serial Phantasm or SE.RA.PH, which handles the summoning of Servants. Servant data is taken from the Automaton (Moon Cell Automaton), a spiritual computer system built on the lunar surface by a pre-human civilization and discovered by humans in the year 2032.
The Automaton has monitored and recorded all occurrences on the Earth (including the activities of human heroes) since time immemorial. In this sense, it is similar to the Throne of Heroes, but it's possible to summons beings which aren't present in the Throne.

Each of the 128 participants in the war must duel the other Masters in a place called the Arena, where failure results in deletion from both the real and digital worlds. When only one remains, the "Holy Grail" shall grant the victor's wish.