Is it possible to run a Docker container without an image?

It is impossible to run a Docker container without an image. An image is a template from which a container is created. A container is a running instance of an image. Without an image, there is simply nothing to run.

You can create an image from a Dockerfile or use an existing one, but the container itself always relies on an image. Even if you create a container with an empty filesystem, it is still based on some base image, even if minimal.