Surprisingly, this is a question I get a lot: do I (or can I) wear ballet shoes in barre class?
However, the floor is either a thin carpet (ugh, Pure Barre, Dailey Method, why???), or more likely, a smooth hardwood floor that is great for a fitness class, but not so much for ballet shoes, which will slip.
Barre class takes place on the smooth floor for most of the class, and an exercise mat for part of it.
You will never do turns or pirouettes in a barre class (unless it is a ballet barre class in a ballet studio, but really, that is unlikely).