1. A carbon-fiber-precursor acryl fiber bundle comprising a polyacrylonitrile-based copolymer comprising 95 to 99 mol % of an acrylonitrile unit and 1 to 5 mol % of a hydroxyalkyl (meth)acrylate unit, wherein the fiber bundle has a single-fiber fineness of 1.5 dtex or more and 5.0 dtex or less and a roundness of 0.9 or less in a cross-section shape perpendicular to a fiber axis of the single fiber, the roundness is a value determined using equation (1): roundness=4πS/L 2 , where wherein S is a cross-sectional area of the single fiber and L is a circumferential length of the single fiber, and S and L are obtained by observing, under an SEM, the cross-section of the single fiber perpendicular to the fiber axis of the single fiber and analyzing the obtained image.