You can easily set up a server on a system such as an old computer for your home network or you can get some of the turnkey nas'es (networked addressed storage or network hard drive that have a server built in). Some of them even support what is know as the (M/L/W)amp stack. If you have one of these units there are a few starter programs that can make access to your personal media very easy.
For pictures, you might use Gallery2 (gallery2.org). Some network hardrives come with this software now. It is what we use to store pictures and makes it real easy to access them with just a web browser.
If you want to easily access your media then try xbmc (xbmc.org) which is available for the major platforms and does not need a web server to work.
For the web server there are a host of programs to look at such as Jinsora, Plex, Mediatomb and etc.
If you just want to server only music, you might want to consider MT-daapd also known as Firefly (will run without a web server). Most itunes compatible devices should connect to it.
You can even set up your own private blog and include media. Wordpress (www.wordpress.org) is probably one of the more popular ones.
The home web server is the next home appliance in my opinion. Have been using one for quite a few years. (http://www.instructables.com/id/Uses-for-your-own-private-cloud/)
Gave you just enough to whet your appetite.