Where are Drupal8 contact forms entries stored?

Contact Storage module will provide storage for Contact messages which are fully-fledged entities in Drupal 8.

This plus core contact module aim to provide functionality equivalent to the base-features of Webform or Entity Form.

Contact Storage

https://www.drupal.org/project/contact_storage

Contact Storage module provides storage for all the messages submitted through the forms on your site as well as views integration to manage them.

The admin view is configured with the ability to look at all of the submissions at once or filter them by individual forms.

Configuring the site-wide Contact form

• Navigate to the Modules administration page Menu > Extend 
• In the Core module section, enable the Contact module. Click Save configuration.
• Navigate to the Contact Form administration page (Menu > Structure > Contact forms)