I tried using Google Drive to sync it up, but Drive is useless for this - it doesn't open the file using the right intent on Android ("file type not recognised" or something similar it says, this used to work as well) and the Drive website makes it a pain to upload an updated file even from the desktop using Chrome.
In my case, I use KeePass 2 and KeePass2Android with Google Sync and it works decently well (I would recommend you try this). I would never recommend it to non-technically-minded folks though.
Sync looks to be for Google-domains/business only. In fact Wikipedia says it has been discontinued! I used to sync over owncloud and that worked pretty well, but the provider shut down and I haven't gotten round to setting another up.
If you're on Android, Keepass2Android [1] is an excellent app that implements the input with a special keyboard. This avoids risking your password via the clipboard. It even comes with a no-network-permission version!
[1] https://play.google.com/store/apps/details?id=keepass2androi...