As soon as you set it up so that 2 or more computers can connect to each other, you have a rudimentary network.
Anyway, a client/server environment is used so that the work done by one can be seen by all/others, e.g. you input data into a database on the server. Others can run reports from that database and see the work you have done. Or, they can put in additional data into the same database. If the database did not reside on the server, then when you put data into the database, you would have to share or copy that to the other person putting in data. This would be doubling the efforts.
For example, if I access a spreadsheet (which obviously is not a database for the picky techies out there, but serves to demonstrate the point), then when I close it, you can see what I have done and may add to it and I will be able to see what you have done.
Additionally, if you built aclient/server application that did not reside on the server, then you would not be able to test it to see if it actually works the way it is supposed to. Testing is HUGE.
Moreso, by making something client/server means you do not have to install everything on each machine. You do not have to worry about what "version" each user is seeing.
Although, you can have client/server architecture without a de facto server. You would have to set one machine as the repository (where the actual file is stored) even though it is someone's workstation, and make it visible to any of the other computers you want to let have access to it. However, you get into slowed response on the repository machine and/or sharing violations.
Anyway you slice it, you cannot have client/server without a network. Even if it is just 2 machines talking to each other, it's a network.