To the northeast of Amman there is a station for Steam Locomotives. This rail way was built in 1909 to connect between Damascus in Syria and Mecca in Saudi Arabia thorough Jordan( in those days Mecca was part of Hijaz ). But unfortunately it was destroyed during the World War 1 and split into many short rails.
If you want to enjoy slow train travel, you can go to Damascus from Amman. Fee is very cheap. From Maan ( Town 200km south of Amman ) to Aqaba ( on the Red sea ) train connection is also available.
I hope the rail will be repaired and we can travel all of the original distance someday.