"You can download all of your saved search history to see a list of the terms you've searched for. This gives you access to your data when and where you want," informs Google. "When you download your past searches, a copy of your history will be saved securely to the Takeout folder in Google Drive. You can download the files to your computer if you want a copy on your computer."
Google will send you an email when your archive is ready to download.
The download dialog is pretty unusual. It includes a warning message: "Please read this carefully, it's not the usual yada yada." It suggests users to enable 2-step verification and it informs them that the archive includes sensitive data. "If you have decided to take your data elsewhere, please research the data export policies of your destination. Otherwise, if you ever want to leave the service, you may have to leave your data behind."
Here's the email message you'll receive when "your Google search history archive is ready".
You'll get a ZIP archive with a lot of JSON files: