@duerer is absolutely right.
The only thing that can change the size of your object with Object Offset is when the Scale of your object or of the offset object is not 1 in x, y and z.

If you make sure that the origin of your object and of the offset object are in the same place you should get something like this:

Personally I would not use Object Offset in the Array Modifier for a wheel or tyre, because you’ll have to make a tapered (and curved) section in order to not get holes (like in above picture) or overlaps and that is harder to model than something straight. Also making adjustments to that section later on is a lot easier with a straight model. That is why I always use an Array with Relative Offset followed by a Simple Deform Modifier set to Bend (360° Angle):

You’ll have to put your Empty at the ‘bottom’ of the wheel, but no need to calculate the rotation of an Empty like in the Object Offset;)
Modifiers can be tricky (you have to get everything correct), but once you’ve figured them out, they are like a found treasure and you’ll be using them all the time!
If you still can’t get it to work, feel free to post your .blend, so that we can see where the problem is exactly.