If you want to rent a storage unit and are looking at sizes that are just big enough to fit all of your stuff in one big pile, you may want to look at bigger sizes instead. While it might seem like a luxury to have a storage unit big enough to allow all those boxes to be arranged in rows with aisles that allow you to access everything, it's really about the safety and well-being of everything in your storage unit. Obviously, if you're throwing things in a storage unit because of something like an emergency move, you might have to go with what's available at the time. But once you get a chance to organize the unit, moving up to a bigger size can have some major advantages.
Letting air circulate around the boxes is necessary to prevent mildewing. You don't need a lot of space around the boxes -- you just need to get them a few inches away from the wall, placed on shelves so they're not stacked up, and so on -- but if you've got everything placed in a pile in a crammed unit, chances are some of your stuff isn't getting the air circulation it needs. That might be alright for a few weeks, but if you're storing things long term, you've got to get more space in there, lest you find the boxes on the inside of the pile suddenly compromised by mildew.
This is really where having all that space starts to be crucial -- without the space, you can't access most of your belongings. If it turns out that you have to find something you've put in storage, then you might end up having to move way more out of the unit first (so you can find the item you need) than is really desirable. With everything organized into rows that you can walk past, you can find what you need more easily.
Literal Damage Control
If your belongings are subject to any disasters or other problems (like roof leaks), having that access will also help you evaluate your belongings for damage. Imagine having to remove everything from the unit to check it over instead of being able to walk to the portion of the unit that had to deal with the leak or whatever it was, and inspecting only those boxes. Damage control would be a lot easier.
Talk to the storage companies like South Town Storage in your area about the sizes they offer and the prices. You might be able to get deals by registering online or by agreeing to rent for a minimum number of months.