No one uses the term "老處女" in the US. It looks quite ridiculous in a sentence. If your research paper is in English I think you should reconsider using that term because it`s not socially known in the US. You should consider using "old maid" or "spinster" as you previously stated. As for "沒人要", I think "unwanted" is a good translation for it.
So I would translate the sentence as follows:
When women reach a certain age, they are often socially condemned to take on identities such as "old maid" or "sinster".