Chapter Two: The Weirdo on Maple Street - S1-E2
Factual error: When Chief Hopper and his fellow patrolman are standing over the lake, Hopper has a radio on his belt. The radio is a modern Motorola walkie talkie and at least a decade newer than it should be.
Factual error: The show starts in 1901 with a woman constable working for the Metropolitan Police Service (Scotland Yard). However women didn't join the Metropolitan Police until 1919 (which is also when the first woman's police uniform was created). Prior to that, women volunteered in the Women's Police Service, but that wasn't formed until 1914.
Chapter Seventy-Nine: Graduation - S5-E3
Factual error: Archie revealed that he joined the Army to avoid repeating his senior year and was shipping to Basic Combat Training the following morning. Before leaving he would need to wait for his background and medical checks to be completed (including getting his dropped murder charge reviewed for suitability), which could take weeks. Also, the Army usually requires a diploma or GED to ship to basic. There were brief periods when the Army was accepting dropouts but it was not when this episode took place.