Not used Big Yellow, but have used one of the other national ones. Insurance was a MASSIVE hidden cost, and they wouldn’t quote it almost till I’d signed the contract.
My advice is to keep your eyes peeled for local, independent storage, usually found on trading estates, or farms if your outside of a big city. The cost is approx 1/4 that of the big boys. Though not sure about insurance in that situation.
If you do go for it, make sure you get the smallest unit you can. And, if its like the one I used, you can move between bigger/smaller units. So if you have an inch of spare space in yours, move to a smaller one.