The incident involving the Air Force KF-16 aircraft took place at around 10:04 local time (01:04 GMT) in the city of Pocheon, near the border with North Korea.
South Korea's Air Force said that it was investigating the incident and apologised for the damage, adding it would provide compensation to those affected.
While shells from live firing exercises sometimes land near civilian residences, they rarely cause injuries.
According to local media reports, two people suffered fractures to their necks and shoulders.
A 60-year-old who was driving when the explosion happened had shrapnel lodged in their neck, Yonhap reported.
"I was driving when I heard a 'bang'," they said. "When I woke up, I was in an ambulance."
"Our KF-16 (jet fighter) abnormally dropped eight shells of MK-82 bombs. It landed outside of firing range," said Korea's Air Force in a statement to the BBC.
The military said the pilot of one of the jets inputted the wrong coordinates by mistake, causing the bombs to drop in the civilian community.
Investigators have yet to determine why the second jet dropped its bombs, the military said, adding all live-fire exercises will be suspended.
One church building and houses were also damaged as a result of the incident.
Images published on local media show a broken window of a building and damaged roof of the church.
A local resident told Yonhap that he had been watching television at home when the explosion, which sounded "like a thunderclap", shook the house.
The director of a nearby senior citizen care centre said that the building's windows shattered and one of their teachers was taken to hospital with injuries. While no seniors were hurt, they said, "they were so frightened that we sent them all home".
Pocheon authorities told the BBC earlier on Thursday that residents had been evacuated while a bomb disposal team worked on safely disposing any unexploded bombs.
But authorities later said they did not find any unexploded bombs at the scene, Yonhap reported.
The defence ministry said the training on Thursday was related to a joint drill with US forces.
South Korea and the US are set to run combined drills from March 10 to March 20 - the first since US president Donald Trump's return to the White House. This comes at a time when the two countries are increasingly wary of the growing alliance between North Korea and Russia.
During another joint drill by South Korea and the US in 2022, troops fired a short-range ballistic missile which malfunctioned and crashed on a golf course in the military base. While the warhead did not explode, it still burst into flames and sent panic rippling across residents in the area.
