If a rental property has a detached storage building used by the owner for storage and not the renter living in the house would it still be exempt or would this be considered business purpose and require flood insurance?

The owner will store lawnmowers and other tools used to maintain the house but it will not be used by the renter.

That does not seem to meet the personal, family or household purposes test.

That is what I thought but just to confirm if it was used by the renter for personal storage it would be exempt correct?

I agree with Randy on this one. You have a landlord who is using the storage building to house equipment that will maintain his property. I have a neighbor that does this exact thing, and in my observation, the structure is not used for personal, family, or household purposes. He stores lawn equipment, tools, and other materials he uses to upkeep his properties. In my opinion, this is clearly a business purpose and would not meet the qualifications for the detached structure exemption.

Now, I could see this situation swinging the other way if the circumstances were different (the owner stores personal items in the structure) but that didn't appear to be the case based on your description.

Yes.