Not necessarily.Small groups of people can do most day-to-day tasks such as handling email and sharing files without needing to install a network server computer.
However, a server does offer advantages even with the smallest group, mainly by centralising file storage, printer sharing, backups, security and software updates.
Often the decisive factor is the need for a group of people to be able to share data in a database. This might be data from an accounting package, contact management system or shared address book. In these circumstances a server running a database program becomes essential.