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I'm updating this answer based on the current version. You can export to a file geodatabase that will contain the attachments.
Navigate to the item page of your hosted feature layer.
Find the layer you want to export features and click the small arrow next to the name to reveal a context menu, and export the data to a file geodatabase. You can then download a ZIP file of the geodatabase, that will contain the attachments
Export of attachments is supported through the API only at this time. It was not included in the update this week.
You can navigate to the REST endpoint of your service, like this:
Click Create Replica
Specify the Layers by index number e.g., 0,1,2 (this service above has 5 layers)
Set the boolean values to true.
Sync Model = none
Data Format = FileGDB
Click Create Replica.
This will start a process that will yield a zip file you can download.