What are some of the longest flights without modern navigation systems?
This list of flight endurance record on Wikipedia lists as number 3 an endurance record of 84 hours 32 minutes between May 25th and 28th 1931. It seems reasonable to assume that no modern navigation means were used in 1931. 
However, the flight was mostly local, so navigation was not really required.

The all time record is from Dick Rutan and Jeanna Yeager in 1986. The Rutan Voyager had state-of-the-art navigation on board, including an Omega Navigation System and a GPS receiver. The latter was only functioning for approximately 4 hours per day, because the GPS constellation was not yet complete and many satellites were missing. 

The Double Sunrise flights seem to qualify, with times in the air between 27 and 33 hours.