Synology does pretty much what you described.
From your side (as the NAS owner/admin):
- Create user accounts
- Allow user to have Read/Write access to "Homes" (which is their home folder)
- Allow user to access "File Station" which is the web-based upload/download. You can also enable other protocol (FTP for example) which would be easier if the user need to upload multiple files rather than via the web (1 by 1).
- Enabled "User Home" service
From the user point of view:
- They have username/password (and your NAS website address)
- Login, access the File Station, and click Upload.
And after the user uploaded things, you as the NAS Admin can go under /Home/username and pick up the files you need.
Do also consider OwnCloud first as it is just a software to re-use your existing hardware. Synology means you have to purchase additional hardware which may/may not in your scope.
Synology Live Demo if you want to see what it looks like: http://www.synology.com/en-global/products/dsm_livedemo
I hope this helps.