A family of Microsoft products that enable users to capture, organize, and reuse notes electronically.
As I mentioned I personally did not come across problems with OneNote and my Linux based NAS by Synology but I can well imagine that BerndP might be right.
There should be no problems (apart from being limited to the Office version of OneNote 2016 and not being able to use OneNote for Mac, iOS, Android or the Win 10 App) to store and access the notes on the NAS. All you need is a network share (preferrably linked to a drive letter) and set OneNotes default storage location to that share in the options (Options - Save and Backup).
Existing notebooks in the cloud can be moved by either creating a new empty notebook on your NAS, opening the one from OneDrive and copying or moving section by section using OneNote or by exporting the existing notebook to a ONEPKG file and loading that one into a new notebook (just double click the ONEPKG in Explorer) and when asked create a new notebook on the NAS to import the ONEPKG content into.
Where the problems that Bernd mentioned (corrupted or misplaced sections) COULD occur is when there are concurrent edits to a notebook/section from different devices and those edits have to be synced and merged from the local caches to the section files on the NAS. But again, I already did this and had no problems by now.
Do you want to use a NAS because you want to access the notebooks from different devices or is it merely about storage space? For notebooks I do not want to be on OneDrive I just store them locally on my work PC and include the storage paths in my backup system (to the NAS).
Regarding using additional MS accounts: Best practice (that will work well with OneNote on non Windows devices too) is to still use just one account for OneNote on all devices but to share all notebooks from the other accounts with edit rights to this "main" account. So you do not have to fumble around with different accounts and logins.
Some additional information why this all is not as easy as it should be:
We are not talking about a simple file synchronisation like Dropbox or Google Drive do. OneNote is capable of having multiple users/devices edit the same "file" (files are actually OneNote sections) at the same time which is not allowed by common file systems (if someone opens a file for edit it is locked for other accesses). In addition OneNote is auto saving whenever a change has been made, may it even be very small.
With that in mind imagine this: You have a large notebook (or section) with lots of pictures, PDFs or attached files. You make a little change to a page within that section (e.g. add a word). The WHOLE section file would have to be uploaded to Dropbox, GDrive or whatever cloud system you are using to update it. Not once, after every small edit! What a waste of bandwith and time (btw. this is exactly what is happening when people force OneNote to use Dropbox by saving the notebook files to the local Dropbox folder). In addition concurrent edits on different devices would lead to sync conflicts on the cloud server as different versions of the files arrive. Usually duplicated (or sometimes corrupt) sections would be the result as the server does not know how to merge different changes to the same file.
In conjunction with OneDrive OneNote is not using a file based sync mechanism but another one called Cobalt. This is also used on Sharepoint and based on splitting the files (in the local cache and on the server) in small chunks which can be individually synced.
Dropbox, GDrive, Owncloud and all other cloud services do not support this protocol, that's why you are limited to OneDrive / OneDrive for Business.
When OneNote files are stored locally (or on a NAS) Cobalt can not be used but Microsoft created a "similar" sync system when OneNote was released 14 years ago that works with the Windows file system. And this protocol is the one BerndP knows to create problems on a Linux based NAS, which -- again -- I personally can't confirm but I very much tend to believe him :)