We live in Burgundy (Bourgogne) and it is very quiet on Sundays. A few small towns have their weekly markets on Sundays, but these aren't tourist-oriented markets like the big one in Beaune; rather, it's a place to pick up vegetables, meat, fish, roasted chicken, some cheese, soap, and some inexpensive clothing (much of it from Italy). There might be the occasional brocante or vide grenier scheduled (these are like organized yard sales/garage sales), a few grocery stores and some boulangeries will be open in the morning, closing around noon, and that's about it. Some museums are open. Some restaurants and cafés are open. There's a reason why many McDonalds stores are packed with French families on Sundays -- not much else is open.
When we feel like doing something other than hanging around the house or riding bikes on a Sunday, it's usually outdoor sightseeing, such as you might do at Semur-en-Auxois, Vézelay, Flavigny-sur-Ozerain, the Abby of Fontenay, or just amble around towns like Dijon, Beaune, Tournus, Cluny, or Mâcon to see the sites and inspect ancient structures and other landmarks.
One sight near Mâcon is la Roche de Solutré. If the weather is good when you're there, it's a nice (not too difficult) hike to the top with pretty stunning views of surrounding vineyards, which in October might still be in their autumn colors. There's a prehistory museum at the foot of the rock that I found pretty interesting in the similarity with stone implements, tools, and weapons found in the Americas -- it's just these are from 40,000 years ago or so, rather than 15,000 years ago. The museum is open 7 days a week. Hours are here. The rocks that form Solutré were uplifted and tilted as part the Alps orogeny (mountain building), which caused considerable structural deformation in this part of France
