Questions & Answers about Я не могу достать книгу с верхней полки.
Here достать means to reach and get/take something, especially when it is not easy to reach.
So Я не могу достать книгу is not just I can't take the book. It often suggests I can't reach it well enough to get it down.
That is why достать is a very natural verb in this sentence.
Because мочь and уметь mean different kinds of can.
- мочь = to be able to, because circumstances allow it
- уметь = to know how to, to have the skill
In this sentence, the speaker does not have a skill problem. They know perfectly well how to take a book. The problem is that they cannot physically reach it.
So Я не могу достать... is correct.