The "flatness" in question isn't the flatness associated with say a piece of paper, it is a description of the overall curve of space-time. With any curvature, you would still have 3 dimensions to move in and objects could be anywhere in that 3D space. If the universe is has positive, then space-time can be described as a sort of 4 dimensional sphere, with 3 dimensions of space and one of time. A negatively curves universe would look like a saddle with space-time spreading away from itself. In a "flat" universe, the warping of space-time is virtually non-existent and would more or less look like a regular 3D space.