HubSpot has silently introduced options that allow for exporting notes more easily:
Navigate to the contact index page (Menu > Contacts > Contacts) or any other object's index page
Click Edit columns
Find the Contacts -> Notes column
Confirm, then Export your view
In the export file, you should find a column containing notes, separated by semicolon if there are multiple ones. Individual notes seem to be capped at 500 characters.
If notes are longer than 500 characters, workarounds are the following:
Creating a custom report (Professional or Enterprise required) and export report
There is also a thread on this topic in the HubSpot Ideas section of the community. I'd recommend upvoting this thread and leaving a comment with your use case.
Best regards!
Karsten Köhler HubSpot Freelancer | RevOps & CRM Consultant | Community Hall of Famer
If you're looking for a simpler way where you do not need to work with the API, you can check out Import2 Wizard. You can export any object from HubSpot (including notes) very easily. You also have the option to filter which notes you export, and define which fields you wish to include in the export file: