We introduce the notion of a combinatorial K3 surface. Those form a certain class of type III semistable K3 surfaces and are completely determined by combinatorial data called curve structures. Emphasis is put on degree $2$ combinatorial K3 surfaces, but the approach can be used to study higher degree as well. We describe elementary modifications both in terms of the curve structures as well as on the Picard groups. Together with a description of the nef cone in terms of curve structures, this provides an approach to explicitly computing the Mori fan of the Dolgachev--Nikulin--Voisin family in degree $2$.