A file-sharing and storage service such as Google Drive and Dropbox is not enough because PostgreSQL itself is a server. It needs to be running on a machine that accepts clients' accesses on port 5432 (by default).
The most you can do in this case is to generate a file with the contents of your PostgreSQL database with a dump tool like pg_dump and share this file via Dropbox or Google Drive so that it is loaded by others in your particular PostgreSQL.
Is there a free cloud service database for PostgreSQL?
Try the ElephantSQL , they offer a free plan for small bases with few concurrent accesses.
All major cloud computing services, such as Amazon AWS , Microsoft Azure , and Google Cloud also offer Postgres servers. As in the reply from @ Shura16, I would also like to quote Heroku , mainly because they contribute a lot in the development of PostgreSQL.