Hospital bed? OK, will the hospital bed have a person in it while driving? Can you get by with something smaller like a stretcher? If you need a full hospital bed your going to take some time loading it, maybe a toy hauler trailer would be better, you can drop the rear ramp and roll it on in. You can get those in shorter lower profile for smaller vehicles.

Renting a storage unit will have issues, some don't even have people there all the time so you have to go through an automated gate, but what if the situation your bugging out from has power down to the gate? Any storage facility easy enough for you to break into will also be easy enough for someone else to break in to and take your stuff.