As 'Saving Private Ryan' was being made, times were changing in Hollywood. New technologies were making both the recording and editing processes more sophisticated, and the analogue age was giving away to the digital age. The internet had been born, and the world would never look back. Spielberg would go on to make plenty of films using these new technologies, but not until after his passion project was complete.
All of the editing on 'Saving Private Ryan' was done on a non-digital system. In real terms, that means it was done manually, with tape reels being cut and then glued back together to complete the edit. It's an intensive work of labor that seems almost unthinkable today, when you can just highlight the parts you don't want and select 'cut'. Some more films were made using old school methods after 'Saving Private Ryan'; and the occasional one is still made today as a curiosity; but this was the last movie made using the old-school method to win an Oscar.