Accurate measurement of optical absorption coefficients from photoacoustic imaging (PAI) data would enable direct mapping of molecular concentrations, providing vital clinical insight. The ill-posed nature of the problem of absorption coefficient recovery has prohibited PAI from achieving this goal in living systems due to the domain gap between simulation and experiment. To bridge this gap, we introduce a collection of experimentally well-characterised imaging phantoms and their digital twins. This first-of-a-kind phantom data set enables supervised training of a U-Net on experimental data for pixel-wise estimation of absorption coefficients. We show that training on simulated data results in artefacts and biases in the estimates, reinforcing the existence of a domain gap between simulation and experiment. Training on experimentally acquired data, however, yielded more accurate and robust estimates of optical absorption coefficients. We compare the results to fluence correction with a Monte Carlo model from reference optical properties of the materials, which yields a quantification error of approximately 20%. Application of the trained U-Nets to a blood flow phantom demonstrated spectral biases when training on simulated data, while application to a mouse model highlighted the ability of both learning-based approaches to recover the depth-dependent loss of signal intensity. We demonstrate that training on experimental phantoms can restore the correlation of signal amplitudes measured in depth. While the absolute quantification error remains high and further improvements are needed, our results highlight the promise of deep learning to advance quantitative PAI.