A file storage and sharing service like Google Drive and Dropbox is not enough, as Postgresql itself is a server. It needs to be running on a machine that accepts customers' access at 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 the pg_dump and share this file via Dropbox or Google Drive to be uploaded by others in their particular Postgresql.
Is there any free cloud service database for Postgresql?
Try the Elephantsql, they offer a free plan for small bases and with few competing accesses.
All major cloud computing services, such as Amazon AWS, Microsoft Azure, and Google Cloud also offer Postgres servers. As in the @Shura16 response, I would also like to quote the Heroku, mainly because they contribute a lot in the development of Postgresql.