This paper presents a study of circumstellar debris around Sun-like stars using data from the Herschel DEBRIS Key Programme. DEBRIS is an unbiased survey comprising the nearest ∼90 stars of each spectral type A-M. Analysis of the 275 F-K stars shows that excess emission from a debris disc was detected around 47 stars, giving a detection rate of 17.1 +2.6−2.3  per cent, with lower rates for later spectral types. For each target a blackbody spectrum was fitted to the dust emission to determine its fractional luminosity and temperature. The derived underlying distribution of fractional luminosity versus blackbody radius in the population showed that most detected discs are concentrated at f ∼ 10−5 and at temperatures corresponding to blackbody radii 7–40 au, which scales to ∼40 au for realistic dust properties (similar to the current Kuiper belt). Two outlying populations are also evident; five stars have exceptionally bright emission ( f > 5 × 10−5), and one has unusually hot dust <4 au. The excess emission distributions at all wavelengths were fitted with a steady-state evolution model, showing that these are compatible with all stars being born with a narrow belt that then undergoes collisional grinding. However, the model cannot explain the hot dust systems – likely originating in transient events – and bright emission systems – arising potentially from atypically massive discs or recent stirring. The emission from the present-day Kuiper belt is predicted to be close to the median of the population, suggesting that half of stars have either depleted their Kuiper belts (similar to the Solar system) or had a lower planetesimal formation efficiency.