If the storage area is an enclosed space and you install the attic fan there, the fan will only ventilate the storage space, and this is only if you provide adequate free inlet area for air to enter the storage area and replace the air you are removing.
You may want to consider installing a quiet attic fan (with a guard below so people can't stick their hands into the fan) that pulls air directly from the finished space in your attic. When you run this fan, it is important to open windows so fresh air from outdoors is pulled in through the open windows, through the attic to the fan, then back outdoors.
Another alternative would be to install the fan above the storage space and put in grilles that allow air to freely travel from the finished space to the storage space. This would move the fan and associated noise away from the living space while still allowing the fan to pull air from the living space (if windows are open to let fresh air in).