Another approach is to use HP Microserver. It provides a "real" server with options to use Xeon, ECC RAM, iLO (remote KVM) in a relatively compact and silent package and not very expensive price.

If you want to use self-built server, you might be interested in pikvm project which allows to build a remote KVM using Raspberry Pi. Remote KVM is very useful for home servers: you can put it in the shelf and use it remotely, including things like tinkering with BIOS, reinstalling OS and resetting.

Also do not forget about fire hazards. The thing that works 24/7 can burn your house. Keep that in mind.