What are your performance requirements? You talk about what software needs to be installed. But there is a difference streaming one compressed audio stream (<100 kByte/second), and streaming a half-dozen high-quality 4K movies.
Another question: Why do you need 3-4 SATA disks? How much data do you have? If you have less than a few TB, the most power-efficient solution would be a single SSD. If you can't afford SSDs, or have too much data for one (SSDs now come in sizes up to 16TB), then the most power-efficient solution would be the minimum nuber of large nearline helium-filled drives.
Since a single disk is unsafe (RAID is mandatory today), you'll eventually need at least a second one. But depending on what your write traffic is, and whether it would be admissible for some writes (like the last hour's worth) to vanish in case of a hardware failure, you could significantly reduce power consumption by having only 1 disk that is always on, and once on hour mirror a snapshot of your file system to the second disk. In a nutshell, you can trade energy usage versus availability and reliability of the most recently written data.
I'll give you two examples of relatively low power consumption. My regular home server (now about 5 years old) is a MicroATX motherboard with a 1.8 GHz 32-bit Atom on it, using a motherboard that is selected to not require power-hungry PCIe cards (there are none), and 4 disks (two large spinning data disks, a boot SSD, and an external backup disk that is always on but physically stored in a large and heavy fire safe). A few years ago, I measured average power consumption to be ~35W. That machine can stream data over the internal ethernet at 20-40 MByte/second (most ethernet ports in the house are only 100baseT), which is enough for video. It also does all the other server functions (firewall, router, DNS, DHCP, NTP, NFS, Apache, ...).
The other example that surprised me recently: I had a Raspberry Pi 3 on my electronics workbench, and for fun connected a keyboard and display to it, and discovered that it is capable of running a Xwindows GUI, play music, and display video. The ethernet is a bit anemic (100baseT, connected internally via USB), but good enough for 10MByte/s. I also recently bought a 2TB Seagate USB-connected disk (it was on sale at Costco for $69, a deal too good to pass up). I bet a RPi3 with two USB-connected disks would have enough power to run as a home server at a throughput of several MByte/s. Whether it could ZFS (which would be my #1 choice for a file system with built-in RAID) is a difficult question though; one might have to use a simpler file system (UFS, ext4, ...) with a separate software RAID layer.