"Cannot start compilation: the output path is not specified for module..."

You just have to go to your Module settings > Project and specify a "Project compiler output" and make your modules inherit from project. (For that go to Modules > Paths > Inherit project.

This did the trick for me.

You have to define a path in the "Project compiler output" field in

File > Project Structure... > Project > Project compiler output

This path will be used to store all project compilation results.

I'm answering this so that I can find the solution when I have to google this error again.

Set project compile output path to path_of_the_project_folder/out. That's what is working today. The intellj documentation makes it seem like we can select any folder but that's not the case.

While configuring idea plugin in gradle, you should define output directories as follows.

idea{
    module{
        inheritOutputDirs = false
        outputDir = compileJava.destinationDir
        testOutputDir = compileTestJava.destinationDir
    }
}