the belief that Spartan 117 is actually the last spartan is actually incorrect. if you read Halo The First Strike and Halo The Fall of Reach, you will realize that a handfull of spartans and Dr. Halsey survived the Covenant assault on Reach.
furthermore, spartans never die. they're just M.I.A.
also, we do know exactly when it takes place. the battle of Reach is a grave chapter in the Human Covenant war. when Reach fell, all UNSC ships initiated Cole Protocol which dictated a random slipspace jump in order to prevent the Covenant from following them to Earth. the Pillar of Autumns jump brought it, its crew, and the Master Chief to Installation 004 (a.k.a. Halo)
and then Halo Combat Evolved happened