The rainbow reflection effect describes the broadband spatial separation of wave spectral components caused by a spatially graded array of resonators. Although mainly studied in optics and acoustics, this phenomenon has recently been demonstrated both theoretically and experimentally for water waves travelling through an array of vertical cylinders. The scattering of two-dimensional linear water waves by an array of vertical, surface-piercing barriers is considered here, in which both the submergence and spacing between the barriers are spatially graded. The rainbow reflection effect arises naturally as wave energy temporarily becomes amplified at different locations depending on frequency. Band diagram calculations are used to demonstrate that this is a consequence of the wave gradually slowing down throughout the array. The wave/barriers scattering problem is then augmented by positioning heave-restricted, rectangular floating bodies equipped with a linear damping mechanism between each adjacent pair of barriers. A solution to the resulting boundary value problem is obtained using an integral equation/Galerkin method. Using constrained optimisation, passive rainbow absorbers are designed that achieve near-perfect absorption (i) over a discrete set of frequencies and (ii) over an octave. This suggests potential applications of rainbow absorbers in the design of smart coastal technologies.