That comparison is not valid. Both of your examples you cited, the paper, table, and book cover, are flat, but they are only two-dimensional, so citing them is oversimplifying the picture of what we suspect our Unvierse to be. In geometry, "spaces" can have any dimensionalities and be either flat or curved. In this sense, flatness refers to properties that determine, for example, what the angles of triangles add up to. In a flat Universe, theser angles add up to precisely 180 degrees, and those of rectangles, including squares, add up to precisely 360 degrees. Three- and four-sided figures that we still call triangles and rectangles can be drawn on surfaces of spheres, but their angles add up to more than 180 and 360 degrees, respectively.

Conceptually, there are spaces with these properties, and in fact, many people, including me, suspect that we have our Universe exists in such a space. One of the purposes for which the Wilkinson microwave anisotropy probe (WMAP) was created and put into orbit at enormous expense was to determine whether the three-dimensional space in which the Universe that we perceive is "flat", meaning not curved in that sense. WMAP data have thus far not been able to detect any curvature. This has led many people to conclude that space is flat. Actually, that conclusion is premature. Space may still possess such a curvature so slight that WMAP has thus far not yet yielded data precise enough to enable it to detect any curvature that may exist. I believe there is plausible reason to expect that WMAP will detect some slight curvature after it has accumulated data with small enough uncertainty to enable it to do so. My belief is based on the fact that the integral of the product of two finite variables must be finite, those variables in the present case being the age of the Universe, found by WMAP to be about 13.71 billion years, and the expansion rate, believed to have always been finite, even during the inflation era when it is believed to have been many times the speed of light but always finite.

I hope the above dispels your confusion regarding the meaning of "flatness" when applied to the geometry of space. If it doesn't, I hope you will ask more questions. Hopefully, one of us can resolve your uncertainty.