Something like an Atom based system would probably fit the bill here.
You will still need a fan in there, as a desktop ATX power supply will need a fan inside it, but it will be near on silent when running.
The always on side of things is just an option in the BIOS screen, there will be an option that is marked (usually) as "Restore on A/C power loss", just make sure that is set to always on. That way, as soon as there is power, the system will restart.
Ethernet port shouldn't be much of an issue, it is rare these days to find ANY motherboard that doesn't have onboard ethernet.
The atom boards are good because they are cheap, and would pretty much fulfill everything that you require, they have a couple of SATA ports, so you could have one hard drive and one cd/dvd drive.
You could also use a laptop (2.5") hard drive to keep the size and power use down, bearing in mind that the laptop drives are slower to read off than desktop (3.5") hard drives, however in this case it would only be a difference of 10-15 seconds tops off boot time, and wouldn't be noticed under any other circumstance.
However, if you are looking to make this a reliable system, up your budget a little bit. Spend another 50 bucks on top of that 100, and the difference in quality of parts will make it much better value over the length of the use you get out of it.