There are a few options for this kind of situation. Whatever you bring with you onto Academy grounds you have to be able to carry on I-Day. There have been some infamous examples of people taking that to an extreme (box boy of 2013), but if your essentials are small enough to carry, you can bring them. They will be stored over BCT and then you will have limited storage once Ac Year hits and you have a little more control over your possessions and space. If it it more than can be carried, I have heard of prior enlisted personnel renting storage lockers in Springs, but I don't know quite the specifics on that.
As for staying at the academy, I believe you can stay on base (as many international cadets have not gone home), but most people would advise either seeing if you can go home with a squad mate or hang out with your sponsors. I will always speak highly of the sponsor program. Love mine and they are basically family now. I know that is not everyone's experience, but generally speaking, you should be able to find a place that is much more "homey" to hang out over breaks than just staying in the dorms.
This is not the first time the academy will face situations like this, so there are resources; you may just have to do a little bit of research to seek out those resources. AOCs and AMTs may know some resources or should know where to ask if they don't. Reach out to the chaplains (even if you aren't religious) as they are great resources. The PEEK performance center will also have resources for cadets in all sorts of situations. Ultimately, the academy wants you to succeed and part of that is making sure that you are safe as much as they can. The military is a big family in a way and tends to look out for their own.
