Can I leave my personal belongings in the basement or storage shed?

This is always a bad idea. First of all, there will be no one looking after the items. Tenants do not want your old stuff and may decide to have a garage sale and pocket the money. Water can enter the space from a pipe dripping from condensation, leaks, or other sources and start a jungle of mold. Please do not request to leave items behind, it never ends well.