How do I centralize all my media?

I'm trying to find a way to have my media as well as my data centralized in my Apple ecosystem in my home. This is what I currently own:

iMac 27" Mid 2012

Macbook Pro Retina 15" Mid 2012

iPad 4th Gen

iPhone 5

Apple TV (newest version)

Airport Extreme

Airport Express (2)

What I'm trying to do is have one large storage device to act as storage for my media and as Time Machine backups for both my computers. I'd like to stream movies and music that I have from whatever device I go with at the highest quality possible. Right now, I have my media on my iMac and its annoying that I have to leave it on at all times if I want to stream to my Apple TV. How can accomplish what I mentioned without having to do that?

Some have said to purchase a NAS like Synology and do the initial backups wired then incremental via LAN and I can have all my movies, music, photos on there to share across all my devices. Some have said to get a Mac Mini but unfortunately isn't enough at 1 TB to hold both my media and my data.

What do you guys suggest I do?