Indy says that he and Oxley were obsessed with crystal skulls - in particular the Mitchell-Hedges skull - in college. Mitchell-Hedges claimed that his daughter unearthed the skull during a dig in 1927, but did not actually begin making that claim until 1943. This is well beyond Indy's or Oxley's college years.
The map showing the trip to Peru has Belize instead of British Honduras. The name Belize for the country was adopted in 1973.
The gas masks used by the bio-hazard crew while decontaminating Indy of radiation are M-17 Gas Masks, which were not developed and used until 1959. Additionally, the M-17 mask is not designed for protection against radiation, making its use for radiation clean up incorrect and dangerous.
A Slip 'n Slide is set up at the nuclear test site. Wham-O first introduced Slip 'n Slide in 1961.
Mutt's Harley Davidson is a 2007 FXSTS Softail Springer, with a hydraulic master cylinder for the front disc brake (neither were avaible until 1972), twincam V-twin first introduced in 1999, a fake kickstarter (not used by Harley since c. 1968), a belt-final drive instead of a chain, a non-stock fuel tank, and a steel plate covering the rear disc brake.
It is possible for the film crew to go full Retraux on the bike, as aftermarket parts suppliers do produce reproduction parts for late-model bikes to look like their vintage counterparts. Or they could just get away with building or renting an old Panhead bike anyway.
The troop transport truck used by the Soviets in the jungle chase is an REO M-35, an American truck. Aside from moving the headlights, it's had little if any modification to imitate a Soviet ZiL 131. It also has a 1970s GM style side view mirror.
The Soviet light truck used in the jungle is a 1961 GAZ 69.
The pistols used by Soviet soldiers, KGB agents and Col. Dr. Spalko are Norinco Type 54, Chinese copy of the Tokarev TT-33. They have large grip mounted safeties (mandated by the ATF on imports in 1990, resulting in many Soviet-era TT-33s being drilled and retrofitted with safeties) and have distinct Norinco grips (no CCCP around the star). Soviet troops never used Chinese weaponry, for the simple reason that they could supply their own. Additionally, the TT-33 was removed from Soviet service in 1952 and replaced by the Makarov PM.
The rifles carried by Soviet soldiers are Norinco Type-56 (fixed stock) and 56-1 (folding stock), Chinese copies of the AK-47 and AKS-47. The Type-56 has a smooth stamped receiver that resembles at a glance the AK-47's milled receiver as well as open top sights. Aside from Soviet troops not using Chinese weaponry, no AK-type rifle had a stamped receiver before 1959, and the Chinese in particular did not adopt it until 1963.
The rocket launcher Indy uses is a Type-69, a Chinese copy of the RPG-7, and seems to be able to fire the PG-2 antitank grenade, which only the RPG-2 can do. The RPG-7 was introduced in 1961.
The pistol Mac pulls on Indy and Mutt is a Browning Hi-Power, but has an external extractor not introduced until 1963. It also has the high profile adjustable "beer can" rear sight introduced in 1971. Interestingly, it does have the correct "ring" hammer, which was replaced with a conventional spur hammer in 1972.