Dropbox can indeed be used with an external hard drive (even though I woulnd't recommend it) as long as you can ensure the drive is always connected to your computer while Dropbox is running and it meets our minimum requirements as described below:
If the drive is ever disconnected from the computer while Dropbox is running there is a small chance that the Dropbox software will start deleting files before realizing that the entire drive has been removed.
If that happens, you will be able to restore your files (assuming that they have been succesfully uploaded in the first place) through your Events page for 30 days if you're on the Basic or our Plus plan while this extends to 120 days for our Professional and Business users.
Last but not least, why not use our selective sync feature which is a great way to free up space on a connected computer by unsycning some of the folders you might not be using at the moment?
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