Let's analyze each sentence to understand why the correct answer is correct:

1. Despite of being a spinster, his life was full of adventures.

This sentence is incorrect because the phrase "despite of" is grammatically incorrect. The correct form is simply "despite" without the preposition "of". Additionally, the pronoun "his" does not match "spinster," which refers to a woman.

2. In spites of a spinster, her life was full of adventures.

This sentence is incorrect for two reasons. First, "in spites of" is incorrect; the correct phrase is "in spite of." Second, the phrase "in spite of a spinster" is not grammatically sound, as it lacks the necessary gerund form "being" to make it coherent.

3. In spite of being a spinster, his life was full of adventures.

This sentence has a correct usage of "in spite of being," but it incorrectly uses "his" when referring to "spinster," which is a term for an unmarried woman.

4. Despite being a spinster, her life was full of adventures.

This is the correct sentence. It uses "despite" accurately, and the pronoun "her" correctly refers to "spinster." The phrase "despite being a spinster" is grammatically correct and makes sense in the context.

As indicated by authoritative sources such as the Merriam-Webster Dictionary and Grammarly, "despite" should be used without "of," and the subject's gender and role references should be consistent.