Excluding the expansion of the tire, the volume of space inside the tire would be the same at 70 psi as at 0 psi (relative to atmospheric pressure). However there would be more air in the tire at 70 psi than at 0 psi due to the fact that it is compressed.
If you took 1 cubic foot of air from a compressor at 70 psi and put it into a balloon, that balloon would expand to a much greater volume than 1 cubic foot. The compressor takes a large volume of air and compresses it into a much smaller volume.
I remember studying the change in volume vs pressure in my college engineering days, but far to distant in my memory to remember how to calculate it.
