IMHO a bare-bones PC is a bit over the top for a home setup. Broken laptop is the way to go if you want to get the power requirements and noise down. Really it comes down to whether the broken laptop will do what you need it to and the price is very competitive compared to even the most basic (and slow) VPS. You can also pop in an SSD and the broken-laptop server sings along.

I should define broken - this is where a keyboard or screen has died yet the actual gubbins work as Intel intended. What you don't want is where the CPU has fried the thermal paste and gone hot for an extended time. That won't run good even if 'fixed'.

A broken-laptop-server will also contain battery backup. The power supply is definitely hot-swappable if you happen to have another power adapter for it.

Really there should be some USB-stick version of a common version of Linux that is completely optimised for power management with the intended use being a redundant laptop turned into a home server. It could have wake on lan magic packet in use too, effectively making it a device that only uses power if you actually use the device. This would be a complete contrast to the normal over-engineered server arrangement.