If you run nightly backups to other partitions, you do not even need RAID. An old office PC would do for that. Something dual-cored would be nice. There should be some Linux distros that serve are ready to go.
For remote Access you need a static IP adress or a router that has VPN functionality (like AVM does) alternatively you would need to use a dynamic DNS service to get the current IP of your home network.
Although I have to say: The Ready-to-go NAS boxes are fine. I could play around with a 4 bay and a 2 bay synology. They are fast enough for most things (Plex is probably going to be too much if you transcode on the fly, simple HD streaming is fine though). And their remote access procedure is really nice and painless.
Although the solution with a used PC is better expandable, I would recommend a ready to go NAS if you actually have a budget and do not need stuff like "Oh I want to host small VMs and stuff".
I run a personal owncloud instance for file exchange and foremost for calendar and contact syncing over various devices without google or apple or even microsoft being involved with their cloud (spying) services.
I was thinking of building a nas with old pc hardware I have around the house, I have an z87 i3 system that's not being used a lot.
In regards to data integrity only a small portion of what is being stored is important, around 90GB the rest is videos, music etc and just junk folders i refuse to delete.
I really would like to use FreeNAS or something similar, Just to I can really get to grips with things, I know wendell mentioned in a recent video that there's an up coming video about creating your own cloud storage.
I may wait till the videos out to get some more ideas.
yeah an i3 is totally fine for something like that, it will be a beast for what you're doing. go for it man! freeNAS is awesome and the knowledge you will gain is relevant.
Plus one for anything involving Owncloud, and Plex! Owncloud does run faster with a real database. Postgresql is definitely faster than MySql. Also NGINX is faster, and generally requires less resources than Apache. Owncloud wiki documents setup for basically all of these possibilities, sans Plex.
Yes i am running a setup close to yours Ubuntu server LTS and running a samba share on an old pc. I have a 1ge lan and it is very stable. I am running a raid 1+1 and I have an automated smart tools testing the drives ever week. It emails me if one of the drives starts to fail so I do not get suprize you dives failed. by the way the os is on its on drive away from the share.