Valid use of goto for error management in C?

As a general rule, avoiding goto is a good idea, but the abuses that were prevalent when Dijkstra first wrote 'GOTO Considered Harmful' don't even cross most people's minds as an option these days.

What you outline is a generalizable solution to the error handling problem - it is fine with me as long as it is carefully used.

Your particular example can be simplified as follows (step 1):

int foo(int bar)
{
    int return_value = 0;
    if (!do_something(bar)) {
        goto error_1;
    }
    if (!init_stuff(bar)) {
        goto error_2;
    }
    if (prepare_stuff(bar))
    {
        return_value = do_the_thing(bar);
        cleanup_3();
    }
error_2:
    cleanup_2();
error_1:
    cleanup_1();
    return return_value;
}

Continuing the process:

int foo(int bar)
{
    int return_value = 0;
    if (do_something(bar))
    {   
        if (init_stuff(bar))
        {
            if (prepare_stuff(bar))
            {
                return_value = do_the_thing(bar);
                cleanup_3();
            }
            cleanup_2();
        }
        cleanup_1();
    }
    return return_value;
}

This is, I believe, equivalent to the original code. This looks particularly clean since the original code was itself very clean and well organized. Often, the code fragments are not as tidy as that (though I'd accept an argument that they should be); for example, there is frequently more state to pass to the initialization (setup) routines than shown, and therefore more state to pass to the cleanup routines too.

FWIF, I find the error handling idiom you gave in the question's example to be more readable and easier to understand than any of the alternatives given in the answers so far. While goto is a bad idea in general, it can be useful for error handling when done in a simple and uniform manner. In this situation, even though it's a goto, it's being used in well-defined and more or less structured manner.