I would say Im reading a long story on the kindle. To me, and my fellow English teacher collegues wouldnt use the term - unless qualified like- 'It really is a long book to read. If it is on it's own it tends to mean that the actual physical book is long. Eg.The book is too long to fit on the shelf.
I would only use 'story' if it were, for example, one story within a book, like a book of short stories or a collection of fairytales. 'A long book' is completely normal to me, though 'thick' would also be natural for me, if describing its physical size.
I would suggest that 'tall' is more likely for a book that doesn't fit on a shelf, since 'long' isn't usually used for height. Unless the books are lying down and it's a very narrow shelf!