ALDI – Sheerness – Opening Hours & Store Details ALDI occupies a good position on Pepys Avenue, approximately a 0.20 mile driving distance west from the centre of Sheerness (nearby Sheerness-on-Sea Train Station). This supermarket serves patrons from the areas of Minster, East Church, Mile Town, Sheerness, Halfway Houses, Warden and Queenborough. This page includes information on ALDI Sheerness, including the opening hours, location info or telephone number.