What does battery acid smell like?

Sulfuric acid isn't very volatile and doesn't have a strong smell.

To the extent it has any smell at all, it smells like sulfuric acid. I know that's not terribly helpful, but think about it for a moment and ask yourself how you would describe the smell of cherries to someone who has never smelled them before. I personally would say "they smell like benzaldehyde", but that's not going to help anyone who hasn't smelled benzaldehyde either.

I guess I could vaguely say that "it smells like vinegar, except it's not quite as strong but it is 'sharper' somehow" and if that helps you, then great.