Web hosting out of your house is very 1990s. It depends on your upstream internet connection, and frankly, it's probably more expensive to do that than simply leasing server space somewhere else (like Amazon.)
There are a multitude of uses for a home server - network nerds like running customized firewalls and routers, I use mine as a PLEX server and file server (backing up my laptop, desktop, housemate's laptop, etc.) You could also set it up to do printer sharing, etc. (Maybe enable AirPrint for iOS devices on printers that don't normally support them.) You can also set up personal cloud services for when you travel.
If you're a gamer, it's certainly nice to host your own server for a game you like, so you can play with your friends with mods or house rules.
The big catch is that your hoard of parts may be worse than useless for a server. Either because they're noisy, draw too much power, or don't perform well enough for what you want to do. This is often the case when repurposing older Core 2 Duo machines for servers - maybe they don't have gigabit ethernet, or DDR2 RAM is too expensive, or it's an old gaming rig with an OC'd CPU, and it draws way too much power to be economical.
So what do you have and what do you want to do?