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A computational search for stable structures among both α and β phases of ternary ATB₄borides (A= Mg, Ca, Sr, Ba, Al, Ga, and Zn,Tis3dor4dtransition elements) has been performed. We found that α-ATB₄compounds withA= Mg, Ca, Al, andT = V, Cr, Mn, Fe, Ni, and Co form a family of structurally stable or almost stable materials. These systems are metallic in non-magnetic states and characterized by the formation of the localized molecular-like state of3dtransition metal atom dimers, which leads to the appearance of numerous Van Hove singularities in the electronic spectrum. The closeness of such singularities to the Fermi level can be easily tuned by electron doping. For the atoms in the middle of the3drow (Cr, Mn, and Fe), these singularities led to magnetic instabilities and magnetic ground states with a weakly metallic or semiconducting nature. Such states appear as non-trivial coexistence of the different spin ladders formed by magnetic dimers of3delements. These magnetic states can be characterized as an analog of the spin glass state. Experimental attempts to produce MgFeB₄and associated challenges are discussed, and promising directions for further synthetic studies are formulated.