As far as “basic” shops are concerned, the busiest small town is Pouilly en Auxois, 10 minutes from Beurizot.
There are three bakeries in Pouilly, a good organic grocery shop (“L’épicerie Comptoir“/ opposite the church) and a good butcher in the main street, just before the square coming from Beurizot.
The pastry shop “Au coeur gourmant” is of good quality (main street, opposite the tobacco shop)
To buy wine: “La cave de Pouilly” (on your left at the exit of Pouilly, direction A6). You will be well advised there.
There are 2 “large” supermarkets: Super U (with fuel 24/24) route de Chailly (open from 8.30 am to 7.30 pm) and Carrefour Market (route de Créancey). The latter shop is open from 8am to 8pm and on Sunday mornings until 12.30pm. The butchers in these shops all sell auxois meat which is usually excellent.
Vitteaux is 8 km from Beurizot by the “mountain” road. You will find a few shops but this small town is sorely lacking in activity.