If you think of a human body (or, rather, the surface of the human body) as grosso modo a misshapen sphere, then you can perform a sphere eversion on it. (E.g. puff up it with air to make it truly round, then do a usual sphere eversion.)
But the human body has features on it (face, outlines of muscles, etc.), and I'm not sure how they end up after the eversion.
A key aspect of the sphere eversion, as was noted in comments, is that along the way the surface intersects itself, i.e. parts of the sphere pass through other parts of the sphere, and so on a physical object this would mean tearing, so that's another problem.
Another aspect of the sphere eversion is that, to achieve it, one has to first distort the surface of the sphere to be far from perfectly round --- so this would also have a pretty destructive effect on a stuffed animal or the like.
If you haven't, you should watch Outside In, whose goal is to have you physically picture the way one of the known eversions works --- e.g. it shows how regions of distortion are introduced as part of the method. (The introduction of this paper also has some interesting history and pictures.)
In the end, your question won't have a purely mathematical answer, but I think if you watch Outside In and try to imagine that the starting sphere is a round stuffed animal (like a stuffed pig or something), you will be able to find an answer of your own that will satisfy you.