Raiders of the Lost Ark is set in 1936, but contains a considerable amount of Anachronism Stew.
German soldiers carry MP 40 submachine guns. The MP 40 (also called the Erma/Volmer machine pistol) was designed in 1938 and adopted by Germany in 1940. While it could be claimed they are meant to represent the MP 36, the MP 36 is very different and was never issued in anywhere near the same numbers as the MP 40.
Soldiers who don't carry the MP 40 carry Mauser Kar 98s, which started manufacture in 1935 and is conceivable it would be out of the country as early as 1936, but they would more than likely actually use Mauser Gewehr 98s.
Several soldiers, including Toht, use a P38 pistol, which was adopted in 1938. Though several hammerless prototypes existed in 1935, it is unlikely anyone would actually be using it at that point in development. Interestingly, Lugers are also used, which is accurate, so it's questionable why P38s would be used at all.
Indy has a Browning Hi-Power as his backup pistol. John Browning was designing it when he died in 1925, and FN started manufacturing it in 1935, so it is not impossible he might have one that early, though unlikely. After he loses it in the fire in Marion's bar, he has a different one, made by John Inglis Co. of Canada, which has a distinctive rear sight "hump". Inglis started manufacturing the Hi-Power in 1944. Interestingly, Indy was envisioned carrying a 1911, but because 9mm blanks were the most reliable, he uses a 9mm Hi-Power.
Near the end, Indy threatens to destroy the Ark with a shoulder-fired anti-tank weapon. According to the film's armourer, it is a Type 56, a Chinese copy of the RPG-2, made to look like a WWII German anti-tank rocket launcher, but with a shoulder grip like an M9 Bazooka. Germany did not have an anti-tank rocket launcher in 1936, nor were they even developing such weapons at that time — the Panzerfaust and Panzerschreck came much later during the war.
The truck Indy, Sallah and Sallah's children climb into is a 1949 CitroŽn Type 23.
The man with the eyepatch's motorcycle is a 1946 BSA M20.
The munitions truck that tips over (then explodes) is a 1944 Mercedes L701, with a wartime expedient wooden cab.
The fuel truck that starts leaking at the airfield is a 1938 Morris Commercial CD Zwicky Flight Refueler, which entered service with the RAF in 1937. The troop transport that Marion shoots up is a 1937 Morris Commerical CV.
An ex-British Army World War II-era Morris 10 HP is parked beside the building the truck carrying the Ark is hidden in.
The truck carrying the Ark is a GMC CCKW, modified to look like a Mercedes LG3000 diesel. Though the cab is rebuilt to look like one, it's too tall and too narrow to be a Mercedes, as well as the actual Mercedes being a 6x4 and not having a powered front axle, unlike the CCKW.
The motorcycle-sidecar combo Indy runs into a pond is a Dnepr MT-11, a postwar continuation of the IMZ M-72, a Soviet copy of the BMW R71 (rejected by the Werhmacht in favour of the R75 in 1939, then sold to the Soviets). Unlike the sidevalve R71 and M-72, it used overhead valves, as did the R75.
The car Marcus drives to Iny's house is a 1937 Lasalle (Cadillac's Companion), although the model year for a brand new car starts halfway through the previous year (i.e. 2014 starts June-2013)).
Indiana flies out of San Francisco to Asia is a Short Solent 3, which first flew in 1946. The exact one used was an ex-BOAC owned by Howard Hughes. It is meant to stand in for the Martin M-130 China Clipper, which started service in 1934 and crashed in 1945. It also makes stopovers at Honolulu (Hawaii), Wake Island, Manila (Philippines) and finally arriving at Nepal (possibly Kathmandu). All of these non-stop flight stretches (which range from 2300-3000 miles) are way too long for the Short Solent's maximum range of approximately 1800 miles (the M-130's range was 3200 miles, for reference).
When Indy and Marion travel from Nepal to Egypt, the map of their route is superimposed over a Douglas DC-3. The DC-3 first flew on December 17, 1935, and the first ones were delivered in 1936. Most of the 300-400 DC-3s in service were flying for American, Eastern, United, and TWA airlines in 1936, and it would have been highly unlikely to find one flying between Nepal to Egypt then. It also makes stopovers at Karachi (Pakistan - part of India in 1936), Baghdad (Iraq) and finally arriving in Cairo. With the exception of the Baghdad-Cairo stretch (about 800 miles), all of these flight stretches (which range from 1200-1500 miles) are way too long for the DC-3's maximum range of approximately 1000 miles.
The conning tower of the U-boat that intercepts the ship carrying the Ark reads "26". The U-26 was a Type IA U-boat commissioned in 1936. The hull is incorrect for that type and that sub; it is actually the full size shell of the Type VIIC U-96 used in Das Boot. The Type VIIC went into service in 1940.
Neither the Type IA nor VIIC had hatches large enough to allow the crate carrying the Ark through. The captain also clearly orders the sub to dive ("Tauchen"), however a lack of constant air supply, limited battery capacity and reduced speed meant U-boats only dove for defense, attack or heavy weather. The only way a U-boat could travel below the surface is with the Schnorkel, which was invented during the war.
The globe in Indy's classroom has African countries that did not exist in 1936.
The map showing Indy's plane's trip to Nepal has Thailand written on it. Thailand was called Siam until 1939. The map to Egypt has Jordan, which was called Transjordan until 1949.
German troops are clearly the Deutsches Afrika Korps (DAK), which was not formed until 1941, as they use its symbol (a swastika on the trunk of a palm tree) and wear tropical uniforms intended for DAK use and introduced in 1940. In 1936, they would have worn either the standard wool uniforms or, given the hot weather, model 33 cotton work uniforms.
Captain Kananga's hat bears the insignia of the Yugoslav People's Navy, which was formed in 1945.
In 1936, Egypt was not under British rule but there was a British military presence, primarily to guard the Suez Canal. It would've been impossible for German troops to operate there without the express permission of the Egyptian government or indeed the British government, e. g. to force Egyptians to work on the excavation site. For the same reason, there was no need for the US government to send Indy to stop the Nazis' search for the Ark.
The Golden Gate Bridge appears when Indy departs from San Francisco. The Bridge was not completed until 1937; although construction began in 1933 and was not open to traffic until 1937, it was nearing completion, so it's feasible it could appear almost completed from a distance in 1936.
Marion lights a cigarette with a butane lighter. Butane lighters were developed in the 1950s.