Two-dimensional electron gases (2DEGs) at transition-metal oxide (TMO) interfaces, and boundary states in topological insulators, are being intensively investigated. The former system harbors superconductivity, large magneto-resistance, and ferromagnetism. In the latter, honeycomb-lattice geometry plus bulk spin-orbit interactions lead to topologically protected spin-polarized bands. 2DEGs in TMOs with a honeycomb-like structure could yield new states of matter, but they had not been experimentally realized, yet. We successfully created a 2DEG at the (111) surface of KTaO3, a strong insulator with large spin-orbit coupling. Its confined states form a network of weakly-dispersing electronic gutters with 6-fold symmetry, a topology novel to all known oxide-based 2DEGs. If those pertain to just one Ta-(111) bilayer, model calculations predict that it can be a topological metal. Our findings demonstrate that completely new electronic states, with symmetries not realized in the bulk, can be tailored in oxide surfaces, promising for TMO-based devices.