Battlefield 2042 may be a multiplayer only game, with no single player element, but it definitely isn't without a story - in fact, as confirmed by a cinematic short film that acts as a prelude to the game's story, the upcoming shooter is an indirect sequel of Battlefield 4.
Exodus, a nine minute short film, reveals that Irish from Battlefield 4 survived the events of that game - canonizing one of the endings - and is alive and kicking as the leader of a no-pat refugee ship in 2042. This puts 2042 into the same continuity as Battlefield 3 and 4, and based on the naming similarities we reserve the prediction that it's in continuity with Battlefield 2142 as well.
The short film depicts what seems to be the direct inciting cause of the renewed war between the USA and Russia, albeit fought by proxy armies consisting of no-pat mercenaries. A secret orbital weapons platform deployed by the USA malfunctions and crashes in Doha, and the mere existence of such a superweapon could reignite the war between the two countries.
An unseen character called Oz, another no-pat, deliberately wants to spark that war in order to force the world to "recognize" the no-pats, who he feels are being ignored and forgotten. Of course, he doesn't reveal this anarchic view initially - Oz hired Irish to retrieve intel about the satellite crash site coordinates for a hefty sum, which a no-pat refugee ship like the Exodus direly needs.