The book might have been entitled Tell Me a Story but the tale did not include any information about what plant these white berries might have come from. I tried a little sleuthing but there are simply too many different kinds of plants with white berries and green leaves shaped this way to figure out which one these might be.
What I can tell you about this vintage white berries drawing is that it dates back to around 1888. It is one of the many illustrations from the book that were drawn by Ida Waugh.
This image is copyright free and in the public domain anywhere that extends copyrights 70 years after death or at least 120 years after publication when the original illustrator is unknown.