Drupal - How to output from custom module without rest of theme

The controller does not have to return a render array, it can also provide a symphony response, for example:

use Symfony\Component\HttpFoundation\Response;

$response = new Response();
$response->setContent('<h1>Hello world</h1>');
//$response->headers->set('Content-Type', 'text/xml');
return $response;

This will bypass the Drupal rendering and theming.

Edit:

You can use two approaches to replace your Drupal 7 code:

1) Override page.html.twig and html.html.twig, remove anything out of this templates and put only that in what you want.

2) Render the HTML like you are doing it now in Drupal 7 and use a Symfony response.

Rendering in Drupal 8:

use Symfony\Component\HttpFoundation\Response;

$output = \Drupal::service('renderer')->renderRoot($build);

$response = new Response();
$response->setContent($output);

return $response;

If you utilize the html_response.attachments_processor and renderer services along with BareHtmlPageRenderer you can still get libraries to load in the head tag of an otherwise un-templated page.

use Drupal\Core\Render\BareHtmlPageRenderer;

$attachments = \Drupal::service('html_response.attachments_processor');
$renderer = \Drupal::service('renderer');

$bareHtmlPageRenderer = new BareHtmlPageRenderer($renderer, $attachments);

$response = $bareHtmlPageRenderer->renderBarePage($build, 'Page Title', 'markup');
return $response;

I do not have enough reputation to add a comment on a reply, but the anwser by @Arosboro is almost complete, except the "build variable is in the wrong spot. It must be the last variable, like so:

return $bareHtmlPageRenderer->renderBarePage([], 'Address', 'checkout_address', $build);

This way the twig variables I placed in $build were rendered correctly.

My full code:

$build = [
  '#theme' => 'checkout_address',
  '#content' => [],
  '#edit' => FALSE,
];

...

$attachments = \Drupal::service('html_response.attachments_processor');
$bareHtmlPageRenderer = new BareHtmlPageRenderer($this->renderer, $attachments);

return $bareHtmlPageRenderer->renderBarePage([], 'Address', 'checkout_address', $build);

And the template:

{{ content }}
{% if edit == true %}
    <button class="btn-edit-address">Edit Address</button>
{% endif %}

This is how I got it to work.