The ability to accurately sequence long DNA molecules is important across biology, but existing sequencers are limited in read length and accuracy. Here, we demonstrate a method to leverage short-read sequencing to obtain long and accurate reads. Using droplet microfluidics, we isolate, amplify, fragment and barcode single DNA molecules in aqueous picolitre droplets, allowing the full-length molecules to be sequenced with multi-fold coverage using short-read sequencing. We show that this approach can provide accurate sequences of up to 10 kb, allowing us to identify rare mutations below the detection limit of conventional sequencing and directly link them into haplotypes. This barcoding methodology can be a powerful tool in sequencing heterogeneous populations such as viruses.