Accellerometers won't help much (as you noticed) because they are not global sensors. They drift. To beat this, you need some way of knowing the location of the object relative to another, fixed location. So, here's my question:
What are the constraints of the system? Is the object to be estimated tethered so it cannot move outside that sphere? or is the sphere an actual object?
If the sphere (or even boundaries of the sphere) are known, fixed, or some other way constrained, then you may have hope of fixing the accellerometer's bias, so long as you can detect that the object has actually not moved (e.g., by sensing the boundary).
You may consider putting sensors in the sphere itself. One way of doing this is magnetic (hall-effect) sensors, then making the object mildly magnetic. Then, whichever group of hall-effect sensors is reading the largest magnetic field is likely near the small object. I have successfully done this to track rotations in a wheel by placing a magnet inside the wheel and hall-effect sensors in the wheel well. Simple, and very, very cheap.
If there is no physical sphere, you may have to put all your sensors in the device. Then the problem is actually very hard! If a cheap solution existed for this, then the Army, Firefighters, Police, and all other sorts of disaster rescue people would love to know how you did it. All solutions I'm aware of require extremely expensive accellerometers, coupled with a GPS or camera system. Tough for cheap applications.