Two children were among those evacuated from the building.
A spokeswoman for the East of England Ambulance Service confirmed that four of those checked over by ambulance staff were later taken to West Suffolk Hospital for further treatment.
Fire crews from Sudbury and Long Melford attended the fire as well as a crew from across the border from Halstead in Essex.
Crews used a thermal imaging camera to check the building and used positive pressure ventilation to help clear the smoke.
The fire was extinguished shortly after 8.47am.
The cause of the flat fire has not yet been confirmed.
Jack Owen, mayor of Sudbury and county councillor for the town, said that he was shocked to hear of the fire.