Access=no:
General public access is prohibited. Stronger interdiction than private. Examples: a fully closed road; a restricted military facility.
It is necessary to add additional access (like foot=yes or bicycle=permissive, etc.) to indicate who may use the element, if anyone is allowed. 

This area is not "restricted military facility", but "restricted nature area".

But, as I know about OSM objections to access=no (widely used), therefore I also suggested:
(vehicle=no) + 
(horse=no) + 
(foot=private)


