We present a multiwavelength study of the double radio-relic cluster A1240 at z=0.195. Our Subaru-based weak-lensing analysis detects three mass clumps forming an ~4Mpc filamentary structure elongated in a north-south orientation. The northern (M_200_=2.61_-0.60_^+0.51^x10^14^M_{sun}_) and middle (M_200_=1.09_-0.43_^+0.34^x10^14^M_{sun}_) mass clumps separated by ~1.3Mpc are associated with A1240 and colocated with the X-ray peaks and cluster galaxy overdensities revealed by Chandra and MMT/Hectospec observations, respectively. The southern mass clump (M_200_=1.78_-0.55_^+0.44^x10^14^M_{sun}_), ~1.5Mpc to the south of the middle clump, coincides with the galaxy overdensity in A1237, the A1240 companion cluster at z=0.194. Considering the positions, orientations, and polarization fractions of the double radio relics measured by the LOFAR study, we suggest that A1240 is a postmerger binary system in the returning phase with a time since collision of ~1.7Gyr. With the SDSS DR16 data analysis, we also find that A1240 is embedded in the much larger scale (~80Mpc) filamentary structure whose orientation is in remarkable agreement with the hypothesized merger axis of A1240.