"Obviously this is impossible."
nonsense. gps works and it works very well. i use it when i fly my plane. i use it when i drive my car.
i'm not going to type in what is already well-documented. the wikipedia article (link) is good. each satellite signal places you on the surface of a sphere centred on the satellite. with three signals you have three intersecting spheres, a point. in practice you need four, to know *when* the spheres intersected. if you have more signals (as you usually do) you can do a least-squares fit.
later: if you know the time you can solve three pseudoranges for (x, y, z). if you don't you solvefour pseudoranges for (x, y, z, t). hence the requirement for four satellites to get a 3d fix, and you get accurate time for free. i work with communication systems that use gps for microsecond-accurate time.
still later: the gps navigation signal really is just a sequence of "now...now...now..." signals. the satellite adds information to say who is saying "now" and exactly when "now" is. the receiver calculates where the satellite was when it sent its signals (the satellites send the necessary data in a separate data stream), compares when it received the signals with when they were sent, and calculates its own position.
part of the calibration work the u.s. naval observatory does is determining the positions of the satellites so they can broadcast the necessary data to gps receivers. they use specialized orbital models that are a lot more complicated than sgp-4 and related orbital models folks like nasa use.
some of us really do know how this stuff works.