It's not as though you need to keep your new storage unit presentable, so you won't ever need to quickly tidy it up before your parents come to visit. But you need to keep it tidy and organised, if only for the wellbeing of your possessions and the fact that you might need to find a specific item at some point during the storage period. You won't quite need to tidy it up on a regular basis, but you'll thank yourself for taking the time to properly arrange everything inside the unit.
You'll need boxes and probably quite a few of them. Delicate items can be stored in a plastic tub with a lid, but it's crucial to make sure that everything is bone dry before that lid is closed unless you want a tub full of mildew. Each cardboard box or tub should have a number clearly written on its outward-facing side with a permanent marker.
These numbers will correspond to the inventory list that you'll make. Yes, you could do this with pen and paper, but you don't want that piece of paper to go missing. Create a document in Word or Google Docs which will simply have the box number followed by a list of the items that have been packed inside. The items might be in storage for an extended period of time, and this allows you to remember precisely what is inside each box.
Given the space constraints, those boxes will probably need to be stacked. Obviously, the heaviest boxes will go on the bottom with the lighter boxes on top so that nothing is crushed. Remember to be sure that those clearly marked numbers are facing outwards when you stack.
An old set of shelves can also come in handy for storing the boxes, as this keeps them off the ground. Be sure that the shelves will be capable of carrying the required weight since your items can be damaged in the event of a collapse. You could also just use shelves for the smaller, lighter boxes or the boxes that contain fragile items.
When stacking the boxes, it's important to create an aisle for you to easily access them. The aisle should be wide enough so that you can lift a box from one section without difficulty, and without backing into the stack of boxes behind you.
It's very simple to properly arrange a storage unit, and while it might seem like a bit of effort at first, you'll be thankful that you did, particularly when you're trying to remember which box contains that photo of you as a kid that your mother absolutely has to see this very weekend!