If you need to stand up a web server along with the database, and maybe a few other components, too, look to the XAMPP stack. This all-in-one solution contains MariaDB plus the Apache web server, the PHP runtime, the Mercury SMTP mail server, web-based controls for all the components, and a service manager for the desktop. It even includes OpenSSL for proper https support.
PostgreSQL
Various repackagings of PostgreSQL as a standalone application have come and gone over the years (see this project, for instance), but it takes relatively little work to set up your own standalone PostgreSQL application. Obtain the binaries minus the setup tools, unpack them into a directory, and run initdb to configure the basic setup. You can then use pg_ctl to start and stop the database as needed.
Python developers have a truly slick option for adding a self-contained PostgreSQL instance to an application: pgserver, a pip install-able library that contains a fully standalone instance of PostgreSQL. The entire app, binaries and all, lives in your Python program’s virtual environment. It does add about 30MB to the base footprint of the venv, but the resulting convenience is hard to match.
