Nearly every political jurisdiction in the US – every one that I know of – has adopted building codes that specify minimum requirements for buildings to be used as legal, permanent dwellings. That means that they have to conform to space and size requirements (building lot sizes), and (again, in every case that I know of) they have to have running water and sanitary facilities. This is not to say that they can’t have chemical toilets or holding tanks, for example, in lieu of septic or municipal sewer facilities, but the running water is still a “must have”. (Many temporary encampments are allowed that skirt this requirement, but they also aren’t considered permanent abodes.)
Electricity and heat are not generally required by most building codes, but if they are furnished, then they have to meet strict safety and quality (of construction) guidelines. That means, for example, that you can’t just “run an extension cord” from another building. That would not pass muster and would not be allowed. Likewise, “heat” would require either electricity, a furnace and fuel supply (or woodstove and a pile of wood nearby) and venting of the flue. That, or a pretty elaborate solar system to enable heating of a such a small area.
Surprisingly, perhaps, because others thought that “building on land you don’t own” is a problem, that’s no difficulty at all. You could work out a long term lease for a dollar, and have “rights” to the private property, but it’s very unlikely that you could build anything habitable on it.
In addition to the building codes, a building permit is also a general requirement. If you don’t obtain the building permit and build anyway, even if it is an elaborate (and expensive) two-story home that seems in all respects habitable… you won’t get a certificate of occupancy and you won’t be allowed to stay there. I’ve seen that happen, and I could show you the house – it’s probably worth more than a quarter-million dollars, sitting empty, abandoned and uninhabitable.