A trans-continental railway traverses an entire continent. A prime illustration is the railway linking San Francisco on the West Coast to New York on the East Coast of the United States.
Option A: Yokohama and Vancouver are situated on separate continents and are separated by the Pacific Ocean, thus lacking a rail connection.
Option B: Auckland and Honolulu are located on different islands, precluding a rail connection.
Option C: Correct — The railway connecting San Francisco and New York is recognized as one of the earliest and most renowned trans-continental lines globally.
Option D: London and Aden are not on the same continent and are not connected by rail.
Therefore, the correct option is (C): The railway line between San Francisco and New York.