It sounds like it's desperately needed. Most NAS boxes are as easy as plugging in and connecting directly to. Can even map it to a drive so users can access it like it was a local disk on their PC. It's also possible to give every user an individual account and create a folder only they have access rights to use.
 
There are some problems with your specific scenario that you need to address however. Putting all the data in a RAID 5 NAS means a single point of failure. A lightning strike or power surge can easily take out the entire NAS then they would really be up a creek. They would need a regular cold backup of the NAS to prevent that. A second problem is that you don't want multiple users working from the same file stored on the NAS, because each time they save their progress it will overwrite the other person's saves and data will get lost or corrupted. You'd need to set up access rules so that can't happen but I'm not knowledgeable on the details.
 
Synology is decent, so is QNAP (what I use).

Thanks for the feedback.

Would it make more sense to put an external storage device (SSDs for reliability) on each computer and set up a daily backup of their accounting data?

There's nothing wrong with a centralized NAS for backups. Depending on how large the office is it would probably greatly simplify things (nevermind being far cheaper than buying SSDs for everyone). Just that in your post you stated you wanted to force all the users to keep their data ONLY on the NAS. As long as their data is on both their systems and the NAS then that's more acceptable. Either they can backup their own data or an automated software can do so.
 
A lot of NAS boxes like QNAP allow someone to plug in a USB drive and have a one-touch button backup feature built in. I'd be easy to have someone do a weekly backup of the NAS in that fashion, just make sure they don't leave the drive plugged into the NAS or the wall, as again it'd need to be a cold backup. 
 

Is that $400 before or after buying disks?? 
 
If you need to save some money then the TS-220 version knocks $90 off the price, the tradeoff is that file transfers will be a little longer. http://www.canadacomputer...357&item_id=060048
 
TS-220 Review at SmallNetBuilder (The go-to website for NAS reviews)

The Synology has a built-in Backup Scheduler and has an App called Time Backup
 
https://www.synology.com/en-us/dsm/5.0/data_backup
https://www.synology.com/en-us/dsm/app_packages/TimeBackup
 
I have a DS212j with 2x 2TB Samsung drives and a USB 2TB Seagate backup.  It works very well.

Just about any NAS would be better than what you got now.
I've used Buffalo, Seagate, WD etc and for anyone that's put a PC together they are pretty easy to setup.
They all usually provide a web based GUI like your home router to set everything up and most support Time machines backup too which are great if you also got Macs.
They also seem to all provide many different and multiple ways to backup for the really paranoid. I just use the RAID and assume no more than a single drive will die at same time so I can just replace it and go on with life. Other people will be paranoid and warn you about this more than single drive dying or some other unnoticed corruption going on NAS before you know it so there are also options on newer NAS boxes to hook up say a external USB drive to run another backup to that and also options to backup to another NAS on your network or cloud.
Really can't go wrong just pick a NAS any NAS.