Here is one of the best tools to backup your Microsoft Office 365 emails is the Office 365 Migration Tool. It is the most secure and powerful tool for backup Office 365 mailbox items into CSV, PST, PDF, DOC, MBOX, MSG, HTML, JSON, etc. The tool ensures a smooth working process without any technical expertise. They allow the performing of selective migration of Office 365 mailboxes through the mail filter option. Additionally, it supports all versions of the Windows operating system. You can download the free demo version of the software to analyze the performance. 
However, if you want to convert Office 365 Emails to Gmail then you choose an informative article is  Migrate Office 365 Email To Gmail. Here in this article, you will get the most sufficient and effective manual method to migrate the process. 

Export Office 365 email data: You can use the eDiscovery Export tool to export the email data from Office 365 to a PST file. This can be done by following these steps:
Log in to your Office 365 account and go to the Admin Center.
Click on the "Exchange" option and select "Compliance Management."
Click on "In-Place eDiscovery & Hold" and then click on the "New" option.
Create a new search query and select the email data you want to export.
Click on "Next" and then select the "Export to a PST file" option.
Follow the prompts to complete the export process. 

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If you find the best solution to migrate Office 365 to pst Gmail then I would like to refer Office 365 mail backup tool.  This Tool for Mac is the best solution by experts to backup emails, contacts, calendars & tasks from Office 365 mailbox to PST, OST, MBOX, and more than 12+ file formats on macOS. You can Migrate Office 365 to Zoho Mail, GoDaddy Email, Gmail, iCloud, Yahoo, Yandex, IMAP, Live Exchange, and many more. This software supports all the latest and below versions of Mac OS.