In the Binary Paintshop problem, there are m cars appearing in a sequence of length 2m, with each car occurring twice. Each car needs to be colored with two colors. The goal is to choose for each car, which of its occurrences receives either color, so as to minimize the total number of color changes in the sequence. We show that the Binary Paintshop problem is equivalent (up to constant factors) to the Minimum Uncut problem, under randomized reductions. By derandomizing this reduction for hard instances of the Min Uncut problem arising from the Unique Games Conjecture, we show that the Binary Paintshop problem is ω(1)-hard to approximate (assuming the UGC). This answers an open question from [BEH06,MS09,AH11].