Ah, back when we had sticks and stones.
Before the Internet, we had something called "client/server" architecture and Local Area Networks. If you weren't trying to establish a connection with a server several miles away, these networks worked perfectly fine to accomplish most anything. You could even establish drive letters and use connections to file servers like a remote hard drive, if you wanted to. If you were several miles away, you could use a Wide Area Network to do essentially the same thing, albeit at a slower speed and at greater expense.
The cheap way to talk remotely was to pass information across telephone lines using devices called modems, and if you wanted to rig up something where two computers talked to each other through computer applications, you did it the same way you do it today: by establishing a communications protocol. There's nothing magical at all about that; both sides just have to agree on what all the bytes mean.
From the very early stages of the Internet, there were ways for machines to communicate across it. Web Services are just the latest flavor of the week.