And before you do this and put something that you might want to access tomorrow on your new storage server - MAKE SURE YOU USE A RAID SETUP!
Let's note that this question was about a small embedded device like a Raspberry PI and not a full blow computer with 6-8 parallel IO SATA channels available. Heck, some options just have USB available. So RAID by any means is not an option here. The hardware we're talking about can barely keep up with a single IO channel. 3-5 parallel channels? No way. REAL LIFE file servers of course have dozens of disks and RAID is one of the many features that allows optimal IO and redundancies to be build into the hardware. But not here - not this example. Just running a single small USB stick is about the expectations here.
Also, as other posts here sorta indicates, RAID isn't "backup". Deleting a file still gets deleted. If you accidentally delete a directory full of thousands of files, they too are deleted. RAID or no RAID. Gone. Linux comes with quite a few tools like LVM and Dejavu to deal with this, but again for a small embedded device that seems overdoing/overtaxing the hardware we can get.
Bottom line is, that for REAL file servers you need more hardware than laid out. On those you can do all the stuff suggested here, and a lot more (look at FreeNAS for instance), and you can make it really smooth so you can loose whole hosts/nodes and not loose availability, automatic cloud backup etc. etc. There are appliances today for consumers that do this (most are Linux based) and for people who just want it to work, that's where I'll go. Plug-in and forget. Cheaper than trying to build it yourself, smaller form factor etc.
As to designing your own NAS, there's plenty of options. Gluster is one of them which I mentioned which does striping/mirroring over multiple hosts as well as disks. XFS/GFS have very interesting file system features for file servers, LVM can handle adding/removing capacity to a running system, consistent snapshots, cached block devices etc. etc. - and with a good web frontend you have a very powerful set of servers that can compete with enterprise/cloud solutions you pay a lot of money from. But it will cost you to get there in hardware alone. It is probably cheaper to just use a cloud provider as your long term storage provider and just download/mirror the data you work on temporarily from the backend storage, automatically sync that across devices etc - and no major hardware investments are needed at all.
Finally, RAID isn't always good. It can make performance worse and make recovery quite complex. I've seen lots of systems running in degraded mode using software raid because it wasn't configured to notify anyone, and the admin didn't know how to tell what the state was, meaning it was actually preforming slower than a single disk would. So it's not a simple choice.