This is how I have successfully done this. Please note caution is_required because it is not recommended to use an external hard drive with Dropbox
First create a separate user account on the computer you are using. Make sure to give it administrative privileges. Mount an external drive, be sure it is big enough, on the computer and ensure it has been given a drive letter. Create a folder on the drive (any name, I use Home) Install Dropbox and specify location as <drive letter>/Home. If you only want specific folders backed up on the hard drive use Selective sync.
Let Dropbox do its work - it may take some time. When done delete the temporary user account.
This might help other users who are interested in backing up their Dropbox folder to another drive.
One thing I'd like to point out, the login on the Dropbox desktop application in the new profile would be considered a separate device, so I'd first unlink the account from the profile first, before deleting it completely.
This is to prevent the account, if it's on a Basic plan, reaching the 3 device limit.
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