I want to make the external Id field to be unsearchable from the global serach or sidebar search. Is it possible? If so how?
External Id is a property on any field that can be marked to represent an external id. So as long as that field is searchable and marked as External Id, it will still be enabled to be searched. So as far as I can say there's no way to hide that from search results.
Refer below excerpt from this knowledge article
Additionally, if you have an external ID field, the field becomes searchable in the sidebar search.
Further from the documentation for Searchable Fields by Object in Lightning Experience and Searchable Fields: Custom Object for Salesforce Classic, refer to below excerpt.
All custom auto-number fields and custom fields that are set as an external ID (no need to enter leading zeros)