My motherboard has 4 SATA ports. I have 16 Seagate Barracuda hard drives (3 TB each). If it is possible, how do I install these hard drives in my computer?
Assuming that you have the physical space for 16 3.5" hard drives, you will need either a port multiplier if your board supports it or a add-in SATA host card or cards that support 16 drives. The technical requirements vary based on what your motherboard supports andwhat you are trying to accomplish. Port Multipliers come in different types, but require a motherboard SATA controller that supports them:
Add-in cards come in different varieties with different features. A similar discussion of them can be found here: http://serverfault.com/questions/237687/cheapest-way-to-connect-20-24-sata-ii-hdds-in-a-budget-storage-server