A NAS is a home storage server, like Synology that you can use to store images, videos and backups, etc on so you can access them from any computer or device in your home. With a couple of clicks, they can easily run applications like Syncthing or Resilio Sync, which are kinda like Dropbox, except you don’t have to pay Dropbox, you’ll just be storing the files on your own service.
If that’s too much to handle, you can still just store your Keepass file in Dropbox, so that it’s available on all your devices. But in the end you’ll still be storing your personal data on someone else’s harddisk.
So in short, is at easy as using a prefab service? No, you’ll have to invest some time, money, and knowledge yourself. But in the end, your data is not gathered in silo together with countless other users, which makes it a lot less attractive for hackers to try and steal it.