There has been growing interest in searching for new low-dimensional (low-D) materials for nanoelectronics and energy applications. Most materials have their structural units extended in three dimensions and connected with chemical bonds. When the dimension is reduced, these bonds will be broken, decreasing the stability and making their experimental realization difficult. Here, we show that stable low-D materials can be made from naturally existing planar structural units. This is demonstrated by first-principles study of boron chalcogenides (B-X), which can have various low-D structures with attractive properties. For example, B2O3 can be the thinnest proton-exchange membrane for fuel cells. B-X are wide-gap semiconductors that can complement the narrow-gap 2D metal dichalcogenides for (opto)electronics. Our work sheds light on the stability of low-D materials and suggests guidelines for rational design of new materials.