There are 3 functions that I can think of in response to your question: export, download and create replica.
1. If you want to use export, you need to be a named user. by default, you can only download your own items or others' items in your organisation if you have the administrator privilege, it is also possible to export others' item if they "allow others to export to different formats". But anyway, I don't think you can export the items in another organisation if you are not a part of that organisation. It sort of make sense that you only can export when you are inside of an org, and when you are outside of an org, you can only "import" its items ...
2. You can download items if they are downloadable, for example, shape files, all sorts of packages (layer, project...), etc.
3. The last one is my favorite, replica. you can checkout feature layers in different formats by creating a replica if the sync is enabled by the service owner. here is an ESRI example how to do it.
Sync overview | ArcGIS for Developers