Even without the sentimental part, cancelling means sorting everything out and finding people to take it away.
Rather than just not paying and letting the owner auction off the contents, maybe you can tell the owner of the storage facility you'd be fine with them auctioning off the contents without ever going into debt?
In theory, one can just hand the keys to a Junk Busters-type company. But yeah - actually doing that is another barrier to action.