There is no feature available to archive issues, unless its the Datacenter hosting option. See https://confluence.atlassian.com/adminjiraserver/archiving-an-issue-968669980.html for more information.
A different way of achieving this is to create a status that comes after the "Done" status of your workflow. Then dont assign the status to any column of the board you are using. In that way you can transition the "Done" tasks to "Archived" and in that way make them disappear from the board (even if they are there, just not visible).
With this solution it is also possible for any persons interested in seeing these tasks to create an additional board that has the status included.
Hope this helps!
Good luck with your Jira endeavors!