Most mechanism to accomplish this motion will require a gear reduction. Since we don't know the motor speed, friction etc. it is not possible to give you a direct answer.
You want to move 300 lbs, 5 feet in one minute (four complete cycles) so the actual calculated power is 1500 lb-ft/minute or roughly .05 hp. The friction in your drive mechanism might be nearly as much. How dirty is the work environment? Does it work continuously? How much room do you have?
Have you looked at a linear stepper motor arrangement?