If an operating system does all the things we know it does, it seems downright impossible for a computer to exist without one.
In reality, the earliest computers didn't have operating systems; they were huge machines tasked with one program at a time. For that reason, they didn't really need operating systems. In fact, the earliest computers required a user to physically connect and disconnect wires from a plug board to retrieve computations. But if you don't have an operating system, can you make your computer do anything?
Yes. But you have a lot of work to do. Without an operating system using and enforcing a standard, systematic approach to running the computer, you're put in the position of writing code (or programs) that must tell the computer exactly what to do. So if you want to type up a document in a word processing program, you'd have to create from scratch code that tells your computer to respond to each character pressed on your keyboard. Then you'd have to write a code that told the computer how those responses must translate to a screen. You'd have to tell your computer how to draw the character you want! Think of every single option or possibility your word processing program has. You'd have to write code for every single one of those directly onto your hard drive.
Let's go back to our general contractor analogy. If we're building a house, we'll want it to have certain features like plumbing, electrical work and windows. In a computer, we also want features like a program that creates documents, one that accesses the Internet and one that stores our photos. Without an operating system, it's not just that your "carpenter" doesn't know where to hammer in nails to a beam to get the room you want -- it's also that you have to forge the hammers, and you have to create the nails.
An operating system provides a uniform set of screws, lumber and any other material you need. It can go back and forth between rooms so fast you didn't even know it left the one you were in.
And that's really important, because here's another thing: Remember how we were talking about the operating system only being able to concentrate on one thing at a time? Well, without one, your computer could run one program. Period. You could create a document. You could save it. You could print it. But you couldn't look at that document and keep a clock running on your desktop. If you don't have an operating system, you're stuck doing one -- and only one -- process at a time.