Is it possible to disable the Node module? Is it possible to disable the Node module in Drupal 7?

As I understand from the interface, it is not possible. Why is the Node module is so important? What if I'm going to use only custom entities?
In Drupal 7 the answer is no, it's a piece of functionality built into the system more than required by anything else. I've heard rumors long ago maybe in Drupal 8 it would not need to be enabled by default -- but I'm not sure of that status now.

It's needed for example because it provides paths to content on the site at node/NID by non-developers -- yes Entities can do that. Drupal caters to a wide range of audiences, not just those that understand what a Node is and what an Entity is. In Drupal 8 there may be an install profile which is super minimal to even not include any content types, I'm not sure what happened regarding that ... ah in Drupal 8 they still require the Node module in the Minimal profile installer:
https://github.com/drupal/drupal/blob/8.0.x/core/profiles/minimal/minimal.info.yml

If you want to use entities then use a module like Rabbit Hole to hide any node content and do not use default nodes at all.