What is the use of a private static variable in Java?

Of course it can be accessed as ClassName.var_name, but only from inside the class in which it is defined – that’s because it is defined as private.

public static or private static variables are often used for constants. For example, many people don’t like to “hard-code” constants in their code; they like to make a public static or private static variable with a meaningful name and use that in their code, which should make the code more readable. (You should also make such constants final).

For example:

public class Example {
    private final static String JDBC_URL = "jdbc:mysql://localhost/shopdb";
    private final static String JDBC_USERNAME = "username";
    private final static String JDBC_PASSWORD = "password";

    public static void main(String[] args) {
        Connection conn = DriverManager.getConnection(JDBC_URL,
                                         JDBC_USERNAME, JDBC_PASSWORD);

        // ...
    }
}

Whether you make it public or private depends on whether you want the variables to be visible outside the class or not.