You can move the Dropbox folder to another drive. If you move it to an external drive, which IS NOT recommended, you need to be absolutely certain that the drive is online and accessible BEFORE Dropbox is started. If the drive isn't available when Dropbox starts, or becomes disconnected while Dropbox is running, there's a chance that Dropbox could see it as a deletion of your files and will sync those deletions to your account online.
For this reason, having the Dropbox folder on an external drive is highly NOT RECOMMENDED.
Why don't use selective sync which is a great way to free up space by syncing only particular folders to a specific device instead?
Anyhow, if you choose to go with the external HDD option, take into account that Dropbox can indeed be used with an external hard drive as long as you can ensure the drive is always connected to your computer while Dropbox (i.e the desktop app) is running. 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.
Additionally, you should ensure that the file system of the external drive meets the file system requirements. Those can be found in the following Help Center article:
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