THE PLT MISTOOK THE TWY FOR THE RWY & BEGAN HIS TAKEOFF ON THE TWY. WHEN HE REALIZED HIS ERROR, HIS SPEED WAS TOO FAST TO NEGOTIATE THE TURN ONTO THE RWY OR STOP & TOO SLOW TO FLY. SUBSEQUENTLY, THE ACFT WENT OFF AN EMBANKMENT ON THE OPPOSITE SIDE OF THE RWY & WAS DAMAGED. THERE WERE 8 PERSONS ABOARD THE 6-PLACE ACFT. THE 2 EXTRA PERSONS WERE NOT USING SEAT BELTS. ONE (A 10 YR OLD BOY) WAS SITTING BETWEEN THE 2 FRONT SEATS WITHOUT A BELT & ANOTHER (A 10 YR OLD GIRL) WAS SITTING ON THE LAP OF ANOTHER PASSENGER. ALSO, SHOULDER HARNESSES WERE AVAILABLE FOR 2 FRONT SEAT PAX, BUT THEY WERE NOT USING THEM. THE PLT HAD BEEN ON AN IRREGULAR SCHEDULE & HAD VERY LITTLE SLEEP IN THE PRECEDING 2 OR 3 DAYS. ALSO, HE WAS RPRTD TO BE UNDER PRESSURE TO RETURN TO NOME TO RESUME A RECORD SETTING AROUND-THE-WORLD FLT INVOLVING HIS 11 YEAR OLD SON. THE TWY ANGLED TO THE APCH END OF RWY 2 FROM THE WEST-SOUTHWEST. THE WIND WAS RPRTD TO BE VARIABLE FROM THE SOUTH AT 10 TO 15 KTS WITH GUSTS TO 25 KTS.