Yes, you can do multiple joins to the UniqueStrings table, but change the order to start with the table you are reporting on and use unique aliases for the joined table. Something like:
SELECT MN.String AS 'MachineName', AN.String as 'AlarmName' FROM Alarm A
JOIN UniqueString MN ON A.MachineName = MN.Id
JOIN UniqueString AN ON A.AlarmName = AN.Id
etc for any other columns