THE PLT USED THE NOSEWHEEL STEERING SYSTEM (NWSS) WHILE TAXIING FM THE FBO TO THE RWY. HE BACK-TAXIED DOWN THE RWY & MADE A 180 DEG TURN, THEN WITHOUT STOPPING, HE BEGAN THE TAKEOFF ROLL. THERE WERE NO RPRTD MALFUNCTIONS OF THE NWSS BEFORE COMMENCING THE TAKEOFF. THE PLT STATED HE HAD DIFFICULTY IN STEERING THE ACFT ON THE TAKEOFF ROLL. AS THE ENGS SPOOLED UP, THE ACFT TRACKED TO THE RGT OF THE RWY CENTERLINE. SUBSEQUENTLY, AS HE CORRECTED WITH THE NWSS, THE ACFT VEERED LEFT, THEN RIGHT, AS IT WAS ACCELERATING. THE PLT INITIATED ABORT PROCEDURES BEFORE REACHING 80 KTS, BUT HE WAS UNABLE TO REGAIN DRCTNL CTL. HE STATED HE RETARDED THE THRUST LEVERS TO IDLE, APPLIED MAX BRAKING & SELECTED REVERSE THRUST, BUT THE ACFT ACCELERATED WHEN THE ENGS SPOOLED UP WITHOUT ACTIVATION OF THE THRUST REVERSERS. HE SHUT DOWN THE ENGS, BUT THE ACFT HIT 3 PARKED ACFT BEFORE STOPPING. THE ACFT WAS EQUIPPED WITH AN EARLY MODEL (DOWTY ROTOL) NWSS. POST-ACDNT EXAMS OF THE NWSS & THRUST REVERSERS REVEALED NO EVIDENCE OF A MECHANICAL MALFUNCTION OR FAILURE.