If we were talking about residential property, the answer is a clear no. They couldn't force you to relocate from an apartment when there is a freeze on evictions unless damage made the unit uninhabitable. For example, if the building suffered a flood or fire so bad it is not legally habitable, they would have to relocate you or help you move immediately.
If the self-storage facility had similar damage, they could argue that. Or they may say we have so much damage we can't protect your property, we recommend you come get it.
I don't think they could argue, "We want to change the doors, gates and aesthetic stuff, move." After all, there are still limits on travel and business. It is probably illegal to ask you to move your stuff when you may not be allowed to drive over, pick things up, and rent at a different facility. Given the Wuhan coronavirus shutdown restrictions, you could legally fight that if you live somewhere still under a near-total shutdown - like Michigan or New York. In this case, do talk to an attorney.