Most file formats (e.g. rvt) are stored directly on OSS, so once you get to that, you can download the file using the GET objectName end point:
The content for Fusion Designs were not exposed for a while, and even though they are available again, in case of complex Fusion models (where multiple models are referencing each other: http://help.autodesk.com/view/fusion360/ENU/?guid=GUID-BAA13D5C-3693-4B9E-9AD0-9842495A3930) you would still need to take care of the references and make sure you can import them back into Fusion 360 later on, with the references automatically resolving.
The best thing for that is to download the Fusion Design in an Archive - and that mechanism will also download the referenced Fusion Designs and have information about the file references.
First you need to call the POST downloads end point and specify f3z (a Fusion Archive zip) file format for download:
This will reply with a body where the data's "type" will be "jobs" - first the "status" will be "queued" then sometime later "processing" and once the download is available the data's "type" in the body will be "downloads" and it will contain a link to the file that you can then download.