Back in the day, we used to use a simple spool. Each number had a tear-away tag on it. As the runners crossed the line, they were put in order (sometimes by physical force) by officials at the finish line. They then entered the chute, narrow enough for only a single file line so nobody could try to move ahead of anyone else. Tags were torn off at the end of the chute and put on the spool in order. Afterward, the tags were numbered in finish order, taken off the spool, and taped to a large table under each team's name. Any team without 5 finishers was eliminated and for the rest, points were added up by hand.
It worked about as fast as the South Regional did it by computer today.