Hey Angus Hooper,
I have been looking for this information for a few clients now and it doesn't look like its a feature. The documentation covers the data in transit (HTTPS) but there is no mention of at rest.
I have found this which may be useful for you. Its for 10.3 but suggests that whole disk encryption should be utilised: ArcGIS for Server 10.3 Security Technical Implementation Guide
Happy to discuss further.
Thanks,
Ben
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