Other than "cheap", there's not much to go on there about your current configuration, longer-term plans or requirements growth... So... In no particular order...
Servers bring with them more complexity in the configuration and setup, if you want to use their capabilities. You'll need to set up DNS and some other services with OS X Server, for instance.
Cheaper and usually less complex than a server box will be a Network Attached Storage (NAS) device, or a (larger) RAID array connected to your existing system — either in place of the server. Synology makes some well-regarded NAS boxes, and there are many other reputable NAS venders.
Four terabytes is one single disk these days. Many several disks, if you want to use RAID storage with your data. (RAID being a mechanism to reduce the changes of data loss due to isolated disk hardware failures. RAID does not protect against volume corruptions, accidental or intentional deletion, theft or other potential causes of data loss.)
Mac Mini with OS X Server, or a Mac Mini Server, does allow an unlimited number of client systems. These Mac OS X Server systems are limited by the available network bandwidth and the physical and software resources available within the server, not by any particular client licensing limits.
Per its licensing, OS X does not allow multiple parallel graphics-based logins into the server box. Remote clients doing file access, mail and calendar are fine, but multiple folks all screen sharing into the server are not.
If you're running Microsoft Windows on your clients systems (and not OS X), you're probably better served with a Windows Server configuration, or with a NAS system. Microsoft Windows can connect to OS X, but managing these two completely different operating systems can be more than many folks want to tangle with unnecessarily.
If you're running OS X and looking to share files, then turn on file sharing in System Preferences > Sharing, and you can then access and use storage on the various OS X systems present.
FWIW and given the tags you've added: Snow Leopard is version 10.6, and very old, and not really where you want to start. Current OS X version is 10.9 Mavericks, and the upcoming release this fall is 10.10 Yosemite.