Assuming you are tech-savvy enough to weigh the pros and cons of the various bits of hardware and software and are comfortable troubleshooting the kind of issues a self-built solution might have, building your own solution for home use is certainly a viable solution. As a bonus, with those skills, you should be able to pin down any relatively common hardware/software issues and fix or replace those yourself whereas on a consumer NAS, you'd have to replace (or RMA) the entire (or almost entire) system.
Given that you're asking the question to begin with, I suspect you may fall short of that level of expertise. Don't let that deter you. If you can afford it as an entertainment expense and don't care if you lose whatever you plan to store there, slapping together a Raspberry Pi, some hard drives, and some software is a good learning experience.
Aside: I think you might find the Raspberry Pi rather slow for large files such as video or audio.
Well, uh, sure, of course you can. If someone else can build it, why shouldn't you (under the assumptions you already made your self)? – Dominik George – 2015-12-24T21:39:25.633
This seems to be asking for an opinion. Not sure if this is really appropriate for superuser. Maybe the Tom's Hardware forums instead? http://www.tomshardware.com/forum/ Or possibly the RPi forums? https://www.raspberrypi.org/forums/
– Ouroborus – 2015-12-24T21:47:40.613You may consider asking at: https://softwarerecs.stackexchange.com/
– kenorb – 2015-12-25T20:09:28.530