Recent studies indicate that eutectic high-entropy alloys can simultaneously possess high strength and high ductility, which have potential industrial applications. The present study focuses on Al0.7CoCrFeNi, a lamellar dual-phase (fcc + B2) precipitation-strengthenable eutectic high entropy alloy. This alloy exhibits an fcc + B2 (B2 with bcc nano-precipitates) microstructure resulting in a combination of the soft and ductile fcc phase together with hard B2 phase. Low temperature annealing leads to the precipitation of ordered L12 intermetallic precipitates within the fcc resulting in enhanced strength. The strengthening contribution due to fine scale L12 is modeled using Orowan dislocation bowing and by-pass mechanism. The alloy was tested under quasi-static (strain-rate = 10-3 s-1) tensile loading and dynamic (strain-rate = 103 s-1) compressive loading. Due to the fine lamellar microstructure with a large number of fcc-bcc interfaces, the alloy show relatively high flow-stresses, ~1400 MPa under quasi-static loading and in excess of 1800 MPa under dynamic loading. Interestingly, the coherent nano-scale L12 precipitate caused a significant rise in the yield strength, without affecting the strain rate sensitivity (SRS) significantly. These lamellar structures had higher work hardening due to their capability for easily storing higher dislocation densities. The back-stresses from the coherent L12 precipitate were insufficient to cause improvement in twin nucleation, owing to elevated twinning stress under quasi-static testing. However, under dynamic testing high density of twins were observed.