"Price: Around 800$ ... need around 16 TB of space. Minimum of 6 TB. "
Disk drives today are around $100 for 2TB drives. 16TB = $800 with no money for the NAS. 6TB @ $300 leaves you $500 for the NAS box.
An OK low end 2-bay (two X 2TB drives = 4TB max) NAS box like this one is $200. plus drives. http://www.newegg.com/Product/Product.aspx?Item=N82E16816111089
A high end NAS box is $400-$500. Plus drives.
A NAS only sends data as fast as the network. Standard 100Mbit wireless network = 10 MB/sec or half the speed of USB. If you are going wireless to and from the NAS cut that in half. For streaming audio/video that's fine. For initially loading the NAS transferring a TB of data at 10Mb/sec takes a LONG LONG time.
A Gbit ethernet wired infrastructure supports up to to 100 MB/sec with good gear, 50 MB/sec with average.
My router lets me attach a USB drive and use it as a NAS device, but I've never tried. Seems bad to load down the router CPU with NAS functions.
You can also get a 6TB (3TB if you raid 1 it) 'personal cloud' lol NAS from Western digital. http://www.wdc.com/en/products/products.aspx?id=620
$420 from amazon, I haven't read reviews. I don't like all-in-ones because when something breaks you throw them away, however they are simple and cheap. Two of them at $840 gives you 12TB of disk (6TB if you raid1 them to protect against drive failure).