it stores your general location within a margin of error in kilometers
Fifty meters to kilometers. Yes. Really accurate. Not.
I worked on this technology for Nortel back in the 90's. Without on board GPS the margin of error depends highly, although not exclusively, on the density and location of the host cells. Another large factor is the type of tower used. Different tower designs move and sway in different ways and that causes a lot of inaccuracy. In urban areas it's generally easier to locate a mobile with a fair degree of accuracy, say a couple of blocks because there is more available receive signal data to use. In rural areas where the cell density is much lower, not so much.