Recently, a class of multifunctional Ti alloys called GUM metals attracts tremendous attentions for their superior mechanical behaviors (high strength, high ductility and superelasticity) and novel physical properties (Invar effect, Elinvar effect and low modulus). The Invar and Elinvar effects are known to originate from structural or magnetic transitions, but none of these transitions were found in the GUM metals. This challenges our fundamental understanding of their physical properties. In this study, we show that the typical GUM metal Ti-23Nb-0.7Ta-2Zr-1.2O (at%) alloy undergoes a strain glass transition, where martensitic nano-domains are frozen gradually over a broad temperature range by random point defects. These nano-domains develop strong texture after cold rolling, which causes the lattice elongation in the rolling direction associated with the transition upon cooling and leads to its Invar effect. Moreover, its Elinvar effect and low modulus can also be explained by the nano-domain structure of strain glass.