Rent a storage unit somewhere. I thought pretty highly of "Door to Door Storage", which rents containerized units out. You can go to their facility on a couple of hours' notice (no charge when I used them) or have the box dropped off in your driveway (additional fee applies, may take several days to schedule).
Your box isn't marked (unless you mark it), so it can't be targeted by thieves. It's in a fairly secure warehouse, piled in stacks with other boxes, so it would be hard to get into anyway. The warehouses have fire suppression sprinklers, too.
Downsides are the slight inconvenience, but that's mostly true of any storage facility. Also the monthly expense. They were charging me $50/month on a longterm plan.
Other storage unit places either give you a little garage-style unit in the open, or a room inside a building. The garage units have a reputation for being robbed and for meth-lab vermin to dump their garbage in the units. The inside-building units tend to be the least convenient in my experience, since you might have to take an elevator to haul stuff in and out; also, although they know who is coming and going from the building, there's no way to prevent someone from breaking into other lockers.