Commonly known for their wine and mustard, this area of France also harvests some terrific single pitch rock climbs! There are roughly 50 different crags around the Bourgogne area, ranging in size from a few routes to hundreds of routes with a full range of difficulties. The crags are mostly limestone, but Vieux-Chateau is granite. And the limestone varies greatly in nature, with some being pocketed a la Frankenjura, some having sharp edges, and some feel like sandstone. If you're visiting Fontainebleau and would like to do some roped climbing without driving too far, this is a good place to visit. The wine here is also terrific, but expensive. For overnighting, you can relatively easily find places to just pitch a tent somewhere, though I'm not sure what the penalty for wild camping is or if it's even allowed. There are plenty of campgrounds around the entire region and they range from very primitive and somewhat horror-film-esque to high-end with a swimming pool! Of course, in all of the sizeable towns/cities like Avallon, Dijon, Beaune, etc. have hotels too.
You need a car because this is a big region. I would recommend a GPS as well, but you can of course get by with a detailed map of the Burgundy region. Dijon is kind of the heart of this climbing area, so it makes a good basis for where the climbing areas generally lie. More detailed descriptions are included in the specific crag descriptions.