Using standard rulers, the surface of the Earth cannot be flat, because it has a non-zero "curvature tensor".
Similarly, space-tme cannot be flat around the Earth, either, because space-time has a non-zero "curvature tensor".
This is assuming that one defines the rather fuzzy English word "flat" with the mathematical defintion "zero Riemann curvature tensor".
The longish quote from Einstein in the thread
may give some insight. This is the quote about rulers on a heated slab, as per the title of the thread above.
For a more mathematical approach that defines what curvature means in 3dimensions, see for instance "Gaussian curvature" in the wikipedia
to be successful at measuring curvature, note that it is assumed that the ants have rulers.