As a business, we carry property and liability insurance as would be expected. However, that doesn't follow through to your contents. The only way we could insure your contents is if we knew everything that was in there and the value of the items. We're not prepared to deal with that, and we're sure you wouldn't be either. As a result, we can't be held liable for any problems that happen to your contents. Unlike some storage businesses, we don't require that you have insurance, but we definitely recommend it, just in case.
Most people will have homeowner's insurance or tenant insurance to cover their contents in your home. That same insurance can normally be extended to the contents in your storage unit -- just talk to your insurance company about it. If your existing insurance doesn't cover the contents in storage, ask them about storage insurance -- that's what they do, so they should be happy to help.