I would go for custom build (PC build) for flexible choice of OS, and what more you needs MySQL.
Also custom build can add-on SATA card later on for more HDD, that's if your chassis will have room for more.

CPU and RAM does affect but if it's home use and less than 10 users, old PC (less than 6 years old is pretty much enough).

Intel C2D is powerful enough to run (Celeron is not recommended for SQL), 4GB RAM or better since you'll be hosting SQL server.
Take note of using 64-bit OS to fully utilize the 4GB RAM.
Also Server class OS (ex. Windows Server) has better network, services performance, cost more (linux free?), but you'll not regret the performance it gives you.

You may need better hard disk than those Eco Green for more than 2 users streaming Full HD movies, need to cater for faster access rather than sequential read. If one user seeking his/her movie (skipping ahead or rewind back), it may be slow or other users may encounter slight pause.
Unless of course you can be sure that all 3 users are accessing from different hard disk.