No matter how careful you are when you are using a self service mover, things can happen. Your uncle could drop a box as he is helping load things into your portable storage container. You could stack a heavy box of books on top of a glass lamp you forgot to label "fragile." Your contents could shift during transport. Or your neighbor could accidentally bump into your storage container while backing out of his garage. Things happen when you move, whether you are having a professional moving company handle everything or doing the move yourself with the help of a self service mover. Protecting your wallet and your belongings in the event that mishaps occur is always a wise decision.
Any reputable self service mover will offer you various insurance plans when you inquire about estimates. Those plans will cover your belongings and the portable storage container during your rental period. The details of the plans are usually listed on the company web site. You can choose a plan that covers only damage to the storage container during your rental period, or a more comprehensive plan that also covers damage to your belongings up to a certain amount. The price of those insurance plans is dependent upon the deductible and plan you choose and is usually purchased on a monthly basis. The self service mover is responsible for damages that occur beyond your control, such as storm damage, vandalism, fire, etc. Make sure you do not pack or store items that are not allowed to be packed or stored in your container (usually included on your rental agreement.)
In addition to insuring your belongings while they are being stored or transported with a self service mover, contact your current renters or homeowners insurance company to see what might be offered. Many renters insurance policies cover your belongings while you move. You can also purchase moving insurance from various companies.
Moving requires a lot of steps, regardless of whether you hire someone to do it all for you or do it yourself with a self service mover. Insurance is always recommended, because things happen. It will not cost you much to provide a safety net for your belongings and your wallet.