Factual error: Druig leads several warriors outside Tenochtitlan as it was sacked by the Spanish conquistadores, and they live peacefully in the nearby forest, for 500 years. The forest is of course the virgin Amazon forest, as captions say. Small problem; Tenochtitlan was in Central Mexico.
Continuity mistake: When Gilgamesh hammers the Deviant into the gates of Babylon, dealing the last strike after the dramatic pose, the shadows cast by the statues don't go much further away than their pedestal. In the following long shot the 5 Eternals have a massive area of shade behind them that extends very far away from the city walls. (00:21:30)
Factual error: There is a billboard in London advertising a display of "artifacts" at the Natural History Museum. British English spells it "artefacts" and the Natural History Museum in London only holds exhibits on natural history. A display of artefacts would be at the British Museum, a completely different institution in a different part of London. (00:07:25)
Factual error: Thena and some Eternals go to the Amazon. The opening scene shows the Amazon River lowlands. In the Amazon they speak Portuguese. But the Eternals speak Spanish to the natives. A part of the Amazon is in Peru where Spanish is spoken, but this is not the area where llamas (shown) typically live. Llamas are highland animals and are in southern Peru, not northern. (01:06:10)
Other mistake: Throughout the movie, the timeframe of Ikaris and Sersi's relationship seems to constantly change; their first kiss is in 575 BC, but Phastus in 2020 asks her how could she stand him for 5000 years. Sprite tells Dane that they broke up 100 years before, but he leaves her because he can't stand to keep secret from her the mission. Which is a problem, since he mentioned that Ajak told him that when they left Babylon, so around 500 BC.
Other mistake: In 5000 BC, Eternals communicate between themselves using contemporary English. That's only expected in a movie, and they speak with natives using their own language, which some of them struggle to learn, nothing strange. Problem is, several Eternals speak English with a very noticeable accent (Scottish, Mexican, Indian) that predates any time when those cultures/nationalities existed. Lauren Ridloff's Makkari being mute speaks with ASL; American sign language, created in the early XIX century.
Other mistake: When Arishem shows his plan to Sersi, the Celestial emerging from Earth is a huge being that completely breaks the planet apart. In the movie finale, the Celestial's hand and head have popped out, but he's much smaller - thankfully, because if he truly had limbs the size of continents, the planet would have exploded before anyone could do anything about it. At that size (again, judging from lack of permanent damage from his half-birth) it's hard to imagine why his birth is supposedly doomsday.
Factual error: The scenes set in Babylon in 575 BCE have the characters speaking Standardized Arabic to one another. At the time in Babylon, however, Aramaic language was actually spoken - Arabic as a language isn't considered by historians to have emerged until the 1st century CE. There's still Aramaic spoken in some regions of Iraq and Turkey, so it's not like the filmmakers have to use Arabic instead.
Plot hole: Spoiler; Ajak and 'the true villain' are the only ones who know the true nature of the mission and the fact that the Earth will cease to exist in 7 days. None of her fellow Eternals would know where to find her or suspect that she's dead or that anything is wrong, but the villain makes them find her body on purpose to provide a distraction to keep them busy investigating her death. Provide a 'distraction' to someone who is completely unsuspecting (and actively lead them) is pure nonsense.
Plot hole: Deviants were created to get rid of dangerous local predators allowing intelligent life to thrive on the planets Arishem 'seeded'. He then created the Eternals to get rid of the Deviants once he realised they eat the species they should have protected. Problem is, it is stated that the Eternals go through their extermination routine multiple times. But the 'mistake' can't be happening all over again in a cycle, and Deviants would ruin a planet if left unattended.
Other mistake: The Eternal's "Domo" is in Iraq. Phastus gives a radar device to Makkari and tells her to go find the Emergence point. We see her sprint away and pass by the Cristo Redentor statue in Rio, and then end up in a spot in the Indian Ocean (she is holding the global map and we see the landmark) that is practically in the Arabian Sea. So the radar brought her literally around the world to get almost to the starting point. Later in the movie Ikaris flies up in space from that spot, and is staring down at South America.
Other mistake: Ajak uses her power to close wounds on other Eternals and when a creature heals itself everyone thinks it is Ajak's power, indicating she is the only one capable of such a feat. However, in the rest of the movie some Eternals pick up significant injuries, but even without 'the healer' are fine.