Learn Croatian: Ovaj sitni zupčanik ne da se izvaditi prstima. - This tiny gear cannot be taken out with fingers.

The construction dati se (literally "to give oneself") followed by an infinitive is a common way to express whether something can or cannot be done. Here, ne da se izvaditi means "it doesn't let itself be taken out" or "it resists being taken out". It emphasizes the physical stubbornness of the object.

Could I say ne može se izvaditi instead? How does the nuance change?

Yes, you could use ne može se izvaditi (it cannot be taken out). However, ne da se adds a specific nuance that the object itself is resisting the action. It sounds like you are actively trying to pull the gear out, but it is physically stuck.

Why is there no preposition like s or sa before prstima?

The word prstima is in the instrumental case, used here to show the tool or means by which an action is performed. In Croatian, when you use a body part or an object as a tool, you use the bare instrumental case without a preposition. Adding s would incorrectly imply accompaniment, as if you were taking it out "in the company of" fingers.

Why use the perfective verb izvaditi here rather than the imperfective vaditi?

You are talking about the successful completion of the action (getting the gear completely out), so the perfective izvaditi is required. If you used the imperfective vaditi, it would focus on the ongoing process of extraction, whereas here the frustration is about not achieving the final result.