We consider the tiny-scale atomic structure (TSAS) on scales of tens of astronomical units, which has been detected by 21 cm absorption lines against quasars with VLBI techniques, against pulsars with time variability and against close binary stars in optical interstellar lines. The TSAS is associated with ordinary cold neutral medium (the CNM). Under the conventional interpretation, the thermal pressure of the TSAS gas is extremely high, some 44 times greater than the Galactic hydrostatic pressure at z = 0 and 300 times greater than the standard CNM thermal pressure. This is unacceptable because the TSAS is quiescent, ubiquitous, and appears to reside in all CNM clouds. Moreover, under the conventional interpretation, H2 should be very abundant in the TSAS, thus exacerbating the pressure problem and also leading to very large extinctions.We consider modifications in the conventional interpretation to ease these dilemmas. They can be eased if the TSAS temperatures are very low, ~7.5 K, but with conventional heating and cooling mechanisms this is impossible without large attenuation of the interstellar radiation field; rather, we estimate the TSAS temperature to be ~15 K. We settle on a geometric solution in which the CNM clouds contain two components: one is the TSAS gas, in the form of either cold, dense curved filaments or sheets; the other is the inter-TSAS gas, which is warmer and less dense. Recognizable TSAS occurs where these TSAS structures happen to line up along the line of sight, producing significant column densities and small transverse length scales even with relatively modest volume densities. We discuss these two-component CNM clouds as complete, integral units and consider the conditions required to simultaneously satisfy the observational constraints on both the standard CNM clouds and also the TSAS. The sheets or filaments occupy a few percent of the CNM cloud volume and contribute about 10% or 30% (respectively) of the total CNM column density. The sheets produce smaller thermal pressures than the cylinders; the sheet pressures can be as small as the standard CNM pressure. We provide suggestions for further observations that will confirm the geometric solution and possibly distinguish between filaments and sheets.