Bari Vecchia is the historic center of Bari. This maze of quintessential Apulian streets and squares is dotted with traditional restaurants, lovely churches, and little moments of everyday life with locals chatting outside their doors.
Yes, it can be crowded at times, depending on when you're visiting, but even in the high season, I've seen this changing - one day can be more hectic while others are tranquil yet lively.
Also, it's not a big area, but take your time to roam it properly to soak in the atmosphere.