It would have been impossible to hijack a train in the Soviet Union and take it all the way to Berlin. The USSR had broad-gauge tracks of 5 ft (1524 mm), while Polish and German tracks are standard gauge, 4 ft 8 1/2 in (1435 mm). While an on-the-fly conversion can be done on passenger cars in more recent times, conversion of the tank wagons in the 1920s would at least have required exchanging the axles using hydraulic devices and would have been impossible to perform on the steam engine - especially at a border post as shown in the movie.
The train hijacked in the Soviet Union on the way to Germany at the beginning of the series is being taken as part of a conspiracy by the Trotskyist Fourth International. This would have been difficult in 1929, as the Fourth International was founded in 1938. Moreover, its members were not in the habit of hijacking trains or engaging in similar acts of what they would have called "adventurism". By the mid-1930s, and especially during the Moscow Trials of 1937, Stalin would be in the habit of charging random individuals with such conspiracies as an excuse to murder them, but these were later proved to be trumped-up charges.
The show is set in 1929. The "Black Reichswehr", a paramilitary sub-organization of the Reichswehr and used in the show throughout season 1 and 2, was disbanded in 1923 after a failed putsch at Kustrin.
In the locomotive scene the Soviet agents disguised as train engine drivers are holding Finnish-made "Suomi m31" sub-machine guns at least 2 years prior they were invented.
First of those guns entered production at 1931-1932, and the particular model shown in the Babylon Berlin is even more goofy: its the war-time muzzle-braked version which was produced from the year 1943- on.
Soviet and German railways have different gauge, hence Russian locomotives and railcars can not pass the border Russia-Germany. Aside from that, there was not one meter of border between the Soviet union and Germany in 1929. Lithuania was between.
*incorrectly regarded as goofs* The letter from Rath's wife refers to Berlin as the capital of the Reich. Most people associate the term 'Reich' as belonging to the Nazi era, when in fact 'Deutches Reich' was the German name for Germany since unification in 1871. The 'Weimar Republic' is a historical designation to describe the post WWI, pre-Nazi era, and not considered an official name.
On several occasions, including the opening sequence, trams are shown with pantographs. Trams in Berlin used trolley poles until 1945, when the network was rebuilt following the war.
This is most certainly due to the production having chartered preserved historic trams in Berlin, which can still run on the city's tram network but modified to use pantographs instead of trolley poles.
Apart from being a 1960s' modernized version of a "war locomotive" type not produced before 1942, the steam engine pulling the tank wagon train also bears an IT-compatible identification number not issued before 1968, with the "-8" at then end being a check digit.
The episode takes place in 1929. The type of steam locomotive shown in the first episode (DR class 52) wasn't built until 1942, and the modernised variant shown was produced in the 1960s.
When we see a computer generated reconstruction of Alexanderplatz in May 1929 it doesn't look anything like the square would have looked at that time.
Construction of the modern building on the right called Berolinahaus didn't start till September 1929.
Construction of the huge modern building on the left called Alexanderhaus didn't begin till 1930.
The company Jonass & Co, who's name we see on the roof of this building didn't move in till 1934.
So the square looks more like it did in the 1930s, except that the square looked totally different by then.
In 1929 these buildings would not have been there, the square would be one huge construction site with still some old buildings left in the middle.
In the first episode Russian train hijackers have a Finnish submachine gun type Suomi KP/31 SJR (a version with a muzzle brake) which was produced after the Winter War (1939-1940). In general we are talking here about a Finnish submachine gun used during WW2.
At the beginning of the first season a train is crossing the German border at Tauroggen from the Soviet Union. Tauroggen (today: Taurage) had been between the two World Wars already a part of the independent Lithuania, as Germany had lost it in WW1, so the German border station couldn't be there in the 20ies (time of the plot). The border at that time was the Memel river near the city of Tilsit. There was no direct border between Germany and the Soviet Union, as Lithuania was in-between.