I will tell you my experience. It may not be your experience.
I had items in a non-climate-controlled storage unit for several years. I didn’t know to wrap them well, and a lot of stuff was in plastic bins or cardboard boxes. The boxes sort of fell apart after a while from heat and humidity. The bins (the cheaper ones) cracked after a while from cold and weight. For paper items, there was a problem with damage from what may have been mold or mildew; it was hard to tell.
It was broken into twice. Since I didn’t have much of anything good in there, they didn’t steal anything and I managed to get two free months of rent from this.
Because a lot of these complexes are in marginal land (like near wetlands, freeway margins, etc.) there is a potential “wildlife” problem. In other words, I had a lot of mouse damage and found a dead mouse or three. These can chew when alive and decompose when dead, ruining or staining what they die upon.
But another problem that people don’t think about when they just want to make stuff go away, is that “away” is convenient but expensive. Unless you are soon to be in a situation when you can remove the items again (say, over the summer while you go home from college, if you’re a student), it becomes easier and easier as time goes by to procrastinate. It’s simpler to pay another month of rent than to clear a space at home and reclaim the stuff. In my case, I actually got the stuff out (ugh) 18 years later when I was appalled by how much money I was spending to store crap ruined by moisture, freezing, and rodents. I eventually threw like 85% of the stuff away or donated what was still usable. So you might end up paying a whole lot of money, more than the stuff is worth. It may be worth considering donating or selling the stuff now, and buying more later (maybe at yard sales) if the items are not particularly sentimental or dear.
Again, I was kind of dumb about this, so if you can learn from my mistakes, you are welcome to do so! In short, I wouldn’t leave anything in one that I wouldn’t want to leave in some kind of unheated, not-too-solid outdoor shed.