Back when token ring was designed normally networks would use coaxial cables for communication. No matter if it ran ethernet, token ring or something else, everybody would share basically a single cable. The cable would have T connectors inserted to connect a computer and the end of the cable needed something to terminate it. It didn't need to be a single line, you could have splits and even a star like design, although there were limitations.

And you are right, any disruption anywhere on the line meant the network would go do. That might be someone removing the termination cap on the end, or simply the line being broken somewhere. However because computers were usually connected using T splitters, it didn't really matter if the computer was connected or not. But the connection not being terminated properly could be an issue. Especially if there was another cable connected to the T before being connected to the computer.

Normally in a room the cable would be laid out like a ring although it usually wouldn't be a closed ring, but instead terminated on one end. This meant each computer would be connected to its direct neighbors, but this wouldn't be an active thing. It wasn't like the computer could only transmit to its neighbors and then they needed to pass it on. It was like a shared line, where everyone could transmit and every computer would receive everything transmitted.

When everything switched over to the regular twisted pair cables we know today, it didn't really change from a communications point of view. Every computer wasn't connected to their neighbors but instead to a hub, but just like before anything anyone transmitted could be received by anyone on the network. It wasn't until much later when things like switches became commonplace and not everyone got all the traffic.