Why run a home server? Why not!
I've run different makeshift home servers for years. Honeypots, web servers, etc. Its really fun! I've learned tons about running Linux in an enterprise and always-online fashion, running different kinds of server software (FTP, SSH, etc), and more.
For hardware, I generally run my servers on various desktops and laptops. One of my favorite computers for running a tiny home honeypot is the Dell Optiplex GX400 (the smallest chassis). It draws almost no power, has a Pentium 4 with Hyper Threading(!), and had room for one hard drive and some DDR2 RAM. While not the most powerful platform ever, it could be stuffed under a desk or on a shelf and was quiet as a mouse.
Later, I moved onto my "halftop". It was a Dell Latitude D620 with the screen ripped off and chassis painted. I had a monitor mounted to my wall, so it was basically a full server/computer in the form factor of a keyboard, since all that was left was the bottom half of the laptop! This was my favorite platform by far for running on my electronics work bench. I could work on Arduino projects with my Arduino IDE and whatnot running on the half-top with the monitor nice and off the desk to give me more room to work!
And finally, most recently, I've been running a Dell Poweredge 2800. This thing is a jet engine. I seriously can't deal with it. Something is wrong with the fan controller, causing the fans to spin at 100% all the time. I worked on fixing it for a while, but quickly realized that even with the dual 604-socket Xeons, the massive 1000w, dual-PSU server had less computing power than my desktop and its AMD FX-8350.
As such, I'm at a crossroads. I have my Poweredge 2800 currently listed on Craigslist, available for a low price. I'm hoping to use the money to pick up some DDR3 ECC RAM, for another sever I was given. This one with an ATX-format SuperMicro motherboard and a nice i5! The only issue is the missing RAM.