Nearly 10% of Earth's continents are covered by river floodplains. These landscapes serve as weathering reactors whereby particles eroded from mountains undergo chemical and physical alteration before being exported to oceans. The time a particle spends in floodplain reservoirs regulates the style and extent of continental chemical weathering and the fate of terrestrial organic carbon. Despite its importance for the global carbon cycle, we still lack a quantitative understanding of floodplain storage timescales. Using a combination of geomorphic mapping, radiocarbon and luminescence dating, and numerical simulations of meander dynamics, we identify well-conserved scaling laws that describe floodplain storage times. Our results reveal that, to first order, floodplain storage durations are set by the ratio of river width to migration rate. The fact that most rivers erode about 1% of their width per year leads to a typical floodplain storage duration of ~5 thousand years.