>>33547701FGO takes place after the Fate VNs, starting in about 2015.
Basically, a magic arctic research team (Chaldea) figured out that the world was gonna end, so they had a plan to send a bunch of talented mages back in time to fix all of the Singularities (weird places in the timeline that started diverging from how things were supposed to be and got all freaky).
But, all of them died because of a weirdo in a tophat was a double agent for the guy who is trying to destroy the world, so they have to send [player insert], a novice dumbass mage, instead. He is joined by Mash (a girl grown in a jar who had her soul bonded to a Servant), Olga Marie (a snooty tsundere and acting director of Chaldea), Romani (Chaldea's doctor and only normal person) and Leonardo Da Vinci, except Leo is a woman for reasons.
It's [your] job to go to all the Singularities and make them stop being goddamn weird by killing some wyverns, flirting with some historical and/or mythological figures and acquiring the holy Grail Fragments that are used to sustain these deviations from the timeline.
Saber and Archer do show up eventually in various forms throughout the story, but Shirou doesn't. Or....he kind of does. It's complicated.