You can definitely use the drives you mentioned in Unraid, I was just pointing out the cost of setting up the number of SATA ports, power supply connections, and appropriate case for 14 SATA drives that you listed would probably exceed the cost of a pair of new drives, 16TB is currently the sweet spot in my opinion for $/GB of storage, You can probably put together something that will work for much less, but it's going to use much more electricity on an ongoing basis, and have many more points of failure than new hardware.
By all means, throw some used parts together and try the trial of Unraid. I'm not being sarcastic, it will probably do exactly what you want for now. I was just trying to prepare you for the inevitable feature creep.
If you put the three 320GB spinner drives in the parity array, and set up two pools, one with the 1TB SSD and another with the 250GB SSD that would be a nice start, you should be able to find an old board with 6 SATA ports. That would give you an entry point to see how Unraid works.