GPS can be used to determine the position with a fair accuracy.
GPS requires you to be able to see the GPS satellites. If you are in a tunnel or on a subway this will not work.
Another way that might be considered to determine your position is using inertial navigation. If you know the position, speed and orientation precisely at a given time and you know all the accelerations and rotations done since that time you can compute the current location.
You need both accellerometers, and gyroscopes to determine your position by this method. Even disregarding budget, the current devices available are not good enough for this for any more than very short periods (seconds).
 is the datasheet for an example of the sort of accellerometer that might be put into the neo.
Accellerometers have a certain amount of noise.
This noise gives you severe positioning errors. An error of only a thousandth of a G gives an uncertainty of velocity of 10mm/s after a second, 1.2m/s after 2 minutes. This is 80m after 1 minute, 8Km after 10 minutes. The integrated gyroscopes are around as bad, and drift typically at degrees per minute.