Achieving small transverse beam emittance is important for high brightness cathodes for free electron lasers and electron diffraction and imaging experiments. Double-gate field emitter arrays with on-chip focussing electrode, operating with electrical switching or near infrared laser excitation, have been studied as cathodes that are competitive with photocathodes excited by ultraviolet lasers, but the experimental demonstration of the low emittance has been elusive. Here we demonstrate this for a field emitter array with an optimized double-gate structure by directly measuring the beam characteristics. Further we show the successful application of the double-gate field emitter array to observe the low-energy electron beam diffraction from suspended graphene in minimal setup. The observed low emittance and long coherence length are in good agreement with theory. These results demonstrate that our all-metal double-gate field emitters are highly promising for applications that demand extremely low-electron bunch-phase space volume and large transverse coherence.