Is there a setting, or a third party application that will allow me to do this? My school gave me unlimited Google Drive storage, and I want to take advantage of it by backing up everything.
To get this result, put the file into a folder within GDrive, wait for it to sync to the server, then set your desktop GDrive settings so that folder is no longer sync'd to your local disk. This will keep the copy on the server and remove the desktop copy.
Once you have GDrive folder(s) that are not sync'd, you can upload more files into it/them via the web page uploader. Alternatively, put files in a sync'd transfer folder, let them sync, then use GDrive's web interface to move them into an unsync'd folder.
Update: The current version of GDrive in 2022 can be configured for either mirroring or streaming. Once you configure GDrive for mirroring, you can right-click any file or folder and pick its "Offline access" setting:
Online only -- You can access this file only when your computer is online. GDrive will fetch it from the server as needed. GDrive needn't keep a local copy when you're not using it, but it has to guess when to discard the local copy.
Available offline -- You can access this file whether your computer is online or offline because GDrive keeps a local copy of it.
Using Google Drive File Stream I had some extremely large files that refused to stay "online only" and would fill my local drive all the way up. I found a trick to fix this. Put these files in a new shared folder using Drive in your browser. Then, in your browser, right click the folder and choose "Hide shared drive". It's still very easy to browse to this folder in your browser, but it'll never try to sync locally.