I use Owncloud for a similar use case. As for Google Drive, OneDrive, ... you need a dedicated app (existing for iOS or Android) or you can access your own Owncloud with the browser. For desktop existing clients for Windows, OSX and Linux.
For installing the Owncloud server on your RPi you need a Webserver with PHP support. If you want to access the files over internet, i'd suggest to use HTTPS. You can use self signed SSL certificates or a SSL cert from a trusted CA.
Here is a tutorial for setting up a Owncloud with "Let's encrypt" certificates
and here a tutorial for self signed certificates
My Owncloud is running with self signed certificates, but the Let's encrypt certificates should be now the preferred solution (nevertheless it's good to learn more about generation of certificates).
In my idea, if you just want to share few files with few persons a SQLite3 data base is sufficient.
As storage you can use a flash drive for storing the files, this will minimize IO on your SD card.