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Call the storage place and ask them. I've done this many times, and they can give you good advice. Also, if you get an idiot when you call the storage place, ask the U-Haul place how big a truck you need for that stuff. Should be the same dimensions you'll need for storage.
I always went a little bigger than I needed, so I could create walk space down the middle so I could get to stuff I wanted (when I did a move like yours), plus it makes it easier to move anyway.