Jedburgh is small and entirely walkable. The abbey, Mary Queen of Scots House, and Castle Jail are all within 10 minutes on foot of each other, and the town centre is easily covered in a morning.
A car is needed for the Border Abbeys circuit — Kelso, Melrose, and Dryburgh are all within 20km but not easily reached without one. The Borders countryside roads are quiet and well-suited to driving.