This sounds simple and effective.
Approximate cost of this set up?
Entirely depends on the amount of data you need saved.
There are different models available which vary in amount of bays (# of drives you can insert) and internal processing power (more power recommended if you choose to have data on your NAS encrypted or want to stream videos)
First off, I would not recommend loading a NAS with drives that are bigger than 4 TB. Even 4 is already borderline too much. I have drives of that size in my RAID 6 setup (8-bay NAS with currently 5 of them filled) and it's taking awfully long to restore the integrity when one drive fails. Now RAID 6 uses two drives exclusively for parity information, so when one drive fails and the system begins restoring the info of the lost drive on a new one, there's still one more drive that may fail (before the restore is complete) before actual data loss occurs. RAID 5 however only uses one drive for parity, and RAID 1 merely means the contents of one drive are cloned to another, so there are less safety nets. You want to keep restore time down in those setups.
So let's say you take 2 TB drives.
If you have a max of 2 TB of data to save, grab a 2-bay NAS model and load it with 2 drives in RAID 1 configuration. Actual data capacity: 2 TB (contents of drive one mirrored on drive two)
For up to 6 TB of data, take a 4-bay model, load it with 4 drives in RAID 5 configuration (one drive used for parity information)
There's also 8-bay models like I have, and even a massive 20-bay one if you need yet more. The bigger the NAS, the more expensive. Synology are readily available on
Amazon, and on Synology's homepage you can get an easy overview and detailed technical comparison of models.
Add the price of the hard drives to the price of whichever model you choose and that's it.
When choosing a model also keep in mind data growth over time, and plan with enough headroom in mind to avoid having to replace your NAS because you are running out of space. You can configure the file sync software to store older versions by the way so you're not screwed if you accidentally overwrite an important file you have no other backups of. But for that you of course need more space.
Late edit: One further addition - include one additional hard drive in your initial purchase plan to have one ready right next to your NAS for when your very first drive fails. When it does, immediately swap it to begin the restore process and order another stand-by replacement.