Strictly speaking you cannot experience story mode via multilayer coop over the network as everyone already mentioned. However there is a mod, which allows to mimic this.
Multiplayer Co-oP Mod allows one to share the same world with anyone connected to the same server. It also allows one to share one's NPCs and traffic. You cannot harm other's NPC though. That said closest one can get to experience story mode on coop is by starting the story mode at the same quest and connect via aforementioned mod's server to share the world, but not sharing NPC in order to avoid doubled NPCs on screen. This will mimic coop play as best as possible. This method has limitations along with mentioned inability to kill other's NPCs and some glitches, but this allows for a feeling like playing in the same world. Also there is a possibility to enter as passenger to other's cars by pressing