In the same quarters - In that place, or that part of the island,
Possessions - Property. His place of residence.
The chief man - Greek: the first man. Probably he was the governor of the island,
Related Commentaries of Acts 28:7
Acts 28:7
7 In the same quarters were possessions of the chief man of the island, whose name was Publius; who received us, and lodged us three days courteously.