As another answer indicated, TRUNCATE TABLE is a fast way to remove the contents of a table, but it's aggressive; it will completely empty the table. Also, there are restrictions on its use; among others, it can't be used on tables which "are referenced by a FOREIGN KEY constraint".
Any more targeted removal of rows will require a DELETE statement with a WHERE clause. Having an index on relevant criteria fields (such as the insertion date) will improve performance of the deletion and might be a good idea (depending on its effect on INSERT and UPDATE statements).
You will need something to "trigger" the DELETE statement (or TRUNCATE statement). As you've suggested, a SQL Server Agent job is an obvious choice, but you are worried about the job being disabled or removed. Any solution will be vulnerable to someone removing your work, but there are more obscure ways to trigger an activity than a job. You could embed the deletion into the insertion process-- either in whatever stored procedure or application code you have, or as an actual table trigger. Both of those methods increase the time required for an INSERT and, because they are not handled out of band by the SQL Server Agent, will require your users to wait slightly longer. If you have the right indexes and the table is reasonably-sized, that might be an acceptable trade-off.
There isn't any other capability that I'm aware of for SQL Server to just start deleting data. There isn't automatic data retention policy enforcement.