A structurable algebra is an algebra with involution \((A,\tau)\) satisfying \((s,x,y)=-(x,s,y)\), \((a,b,c)-(b,a,c)=(c,a,b)-(c,b,a)\) and \(\frac23[[a^ 2,a],b]=(b,a^2,a)-(b,a,a^2)\) for any symmetric elements (with respect to the involution \(\tau\)) \(a\), \(b\), \(c\), skew- symmetric \(s\) and arbitrary \(x\), \(y\) in \(A\). Here \((\cdot,\cdot,\cdot)\) denotes the associator in \(A\). This is equivalent to the original definition by \textit{B. N. Allison} [Math. Ann. 237, 133--156 (1978; Zbl 0368.17001)], but without the assumption of the existence of a unit. On the other hand, an \(H^*\)-algebra is a complex algebra \(A\) with a conjugate linear mapping \(x\mapsto x^*\) and a complete inner product \((.\mid .)\) satisfying \(x^{**}=x\), \((xy)^*=y^*x^*\) and \((xy\mid z)=(x\mid zy^*)=(y\mid x^*z)\) all \(x\), \(y\), \(z\) in \(A\). This paper is devoted to the description of the structurable \(H^*\)-algebras with zero annihilator. To this aim, it is shown how this description can be reduced to the description of the topologically simple \(H^*\)-algebras \((A,\tau)\) such that \(\tau\) is an \(\ast\)-involution, that is, \(\tau\) and \(\ast\) commute. After proving that any semisimple finite-dimensional structurable algebra has a unit, the authors show that every complex finite-dimensional structurable algebra \((A,\tau)\) can be structured as \(H^*\)-algebra in such a way that \(\tau\) becomes isometric. This requires a careful case by case analysis of the finite-dimensional simple structurable algebras. The main result asserts that the topologically simple structurable \(H^*\)-algebras whose involution is a \(\ast\)-involution are the adequate \(H^*\)-versions of the finite-dimensional simple structurable algebras classified by Allison and completed by Smirnov. With this result, the description of the structurable \(H^*\)-algebras with zero annihilator is completed, providing an infinite-dimensional extension of the structure theory of finite-dimensional semisimple structurable algebras.