You can. What it sounds like you're looking for is a distributed file system. That is, one that can take space that is available on each of your machines and pool it into a single filesystem that your server can use.
With Linux on both the desktops and the servers this is fairly easy to accomplish (there is always some setup and configuration with distributed systems). I would recommend GlusterFS. Each desktop on your network would run the GlusterFS service configured to combine their free disk space. You could then mount this on your server and use it for whatever purpose you want. It will be a bit slower than an SSD or HDD but on a local network it shouldn't be too much slower.