In WooCommerce product options edit page display a custom field before the SKU

IS NOT POSSIBLE TO PLACE ANY CUSTOM FIELD BEFORE THE SKU

You can take a look to html-product-data-inventory.php source code file which displays product inventory fields.

But you can display your 'Barcode' custom field just after the SKU field (for example)

For that you have to hook your custom function in woocommerce_product_options_sku action hook instead. There is also some missing things in your code to display a saved value.

And finally you need another function to save that value when product is saved or updated.

Here is that complete code:

add_action('woocommerce_product_options_sku','add_barcode', 10, 0 );
function add_barcode(){
    global $woocommerce, $post;
    // getting the barcode value if exits
    $product_barcode = get_post_meta( $post->ID, '_barcode', true );
    if(! $product_barcode) $product_barcode = '';
    // Displaying the barcode custom field
    woocommerce_wp_text_input( array(
        'id'          => '_barcode',
        'label'       => __('Barcode','woocommerce'),
        'placeholder' => 'Scan Barcode',
        'desc_tip'    => 'true',
        'description' => __('Scan barcode.','woocommerce')
    ), $product_barcode); // <== added "$product_barcode" here to get the value if exist
}
add_action( 'woocommerce_process_product_meta', 'save_barcode', 10, 1 );
function save_barcode( $post_id ){
    $product_barcode_field = $_POST['_barcode'];
    if( !empty( $product_barcode_field ) )
        update_post_meta( $post_id, '_barcode', esc_attr( $product_barcode_field ) );
}

This code goes in function.php file of your active child theme (or theme) or also in any plugin file.

This code is tested and works for WooCommerce version 2.6+ and 3.0+