Personally, I use KDE's file manager (Dolphin). Plus a curl-cased Bash script for when I need to upload an arbitrary file from one of my home computers via my phone (SSH from phone to computer, sometimes over sat link, run "~/bin/upload.sh /some/file").
I manage my own ownCloud servers (one personal, one company), but hosted options are available.
And yes, it does keep file history (optional, enabled by default), as well as encrypted storage (optional, disabled by default).
Neither the client, nor the server even, require any administrative access.