Double negative elastic metamaterials are a kind of novel artificial materials with unique ability to manipulate elastic wave propagation in the subwavelength scale. They are generally designed by inducing the combination of multiple resonance with different negative effective parameters. However, due to the limitation of empirical structural shape and design tools, these metamaterials are made by complex multiphase materials but possess a relatively narrow frequency range with double negativity, which is unsuitable for practical engineering applications. In this work, a shape optimization and single-phase chiral elastic metamaterials (EMMs) based strategy is presented for designing the EMMs with a broadband double negativity (negative mass density and bulk modulus). Several numerical examples are presented to validate the method by considering different initial shapes, target frequency ranges and design variables. Besides, numerical simulations related to double negativity including negative refraction and imaging are investigated by using the optimized chiral EMMs. Interestingly, elastic wave mode conversion and super-resolution imaging of 0.28λ are observed. The proposed metamaterial combined with the design approach is a very efficient way to obtain double negativity over a broad frequency band and it may thus have great potential for designing new elastic metamaterials.