Five-year-old twins Cabell and Ellis Chase are bundles of energy. The dynamos were too busy chasing each other around the house and playing drums on Tuesday to talk about something as mundane as their budding acting careers, but Cabell did take the time out to say he has short lines that are easy to remember and the grownups on the set are nice.

The children of Amy and Robert Chase of Mathews, both boys play the same part—the role of Thomas—in the AMC historical drama series “Turn: Washington’s Spies,” which is about espionage during the Revolutionary War. Thomas is the child of the characters Mary and Abraham Woodhull, who are portrayed in the series by Meegan Warner and Jamie Bell.

Amy said the boys are children first and actors second, and that their participation in the series wasn’t something the family planned.

“It literally fell into our laps,” she said.

The family, including older son Scottie, were residents of Richmond’...