Is it acceptable for a professor to ask post-doc student to babysit for free? Like the title says, I have a friend who is a post-doc and her professor is asking her and another post-doc to baby-sit for him.  He does not pay them, he simply expects them to baby-sit for free because they work for him.  It is not clear whether this happens during the day or in the evening, though by my understanding it makes little difference in academia.  It is also not clear just how common it is for him to ask, but apparently it is at least semi-regular.  To me this seems like an abuse of power and there should be rules against it.  Is this kind of thing normal?
Your friend's advisor has made a very unethical request because, as a supervisor, it makes it harder for the postdocs to feel like they can say no. And they should feel free to say no, since the request is not normally part of any university employment contract I've seen. 

Now, if the request were at the workplace, incidental and of brief duration (something like "Could you watch her while I take this call from the doctor's office?", for instance), then it wouldn't be so problematic (although still less than ideal). But anything more than that—anything that involved a regular arrangement, or was of extended duration—should be handled as a separate business transaction, so as to avoid exactly the coercion problem that you've raised.