Operational Definition of Reference Frame in General Relativity Most treatments of GR begin with the assumption that spacetime is a pseudo-Riemannian manifold (or, sometimes, that it is a more general manifold). But this entails quite a few tacit assumptions about the nature of space and time.
I would like to approach things from a much more fundamental point of view and itemize each and every assumption for pedagogical purposes.
So here is my question. Suppose we are experimenters in a laboratory equipped with various operationally defined instruments. We wish to construct a mapping between points of physical space and $n$-tuples of real numbers ($n = 4$?) such that all the usual properties of spacetime physics hold.
How do we proceed? What measurements and assumptions do we make? What type of manifold results? If it is one requiring an atlas of more than one patch, then what is the answer to the following question? If I am an observer located in one patch, how can I track (via radar?) a particle traveling along a trajectory that goes through all other patches?
 A: Actually, the assumption of a psuedo riemannian manifold doesn't require many tacit assumptions. Can you measure time and distances? Can you define a right angle? Ok, you now have a manifold equipped with a metric. Want to include time as a dimension? Now you have four dimensions. You can't turn around in time like you can in space, so you need the time direction to be different, i.e., the metric has to be $---+$ or $+---$.
As to your question regarding charts, etc., nature does not use coordinate systems. Coordinates are man made things which people use to make sense out of measurements, under some rather "natural" assumptions, like self consistency and the ability of different observers to agree on what they are observing. You are not an observer located in a patch. You are an observer. You assume that others who are ostensibly observing the same thing ought to agree on what it is you are observing. Figuring out how to seperate what you are observing from the measurements you make is what physics is. If there were a well defined prescription for doing that, we would already know everything about the universe.
