The telescopic contact between graphene layers with a dielectric spacer is considered as a new type of graphene-based nanoelectronic devices. The tunneling current through the contacts with and without an argon spacer is calculated as a function of the overlap length, stacking of the graphene layers and voltage applied using non-equilibrium Green function formalism. A negative differential resistance (similar to semiconductor tunnel diode) is found with the peak to valley ratio up to 10 and up to 2 for the contacts without any spacer and with the argon spacer, respectively. The capacitance of the contacts between the graphene layers with the argon spacer is calculated as a function of temperature taking into account the quantum contribution. The related RC time constant is estimated to be about 3 ps, which allows elaboration of fast-response nanoelectronic devices. The possibility of application of the contacts as memory cells is discussed.