Based on observations of the most abundant sulfur-bearing molecules (H2S, CS, NS, SO, H2CS, OCS, and SO2) carried out with the IRAM 30m telescope and SEST, we present the first analysis of the sulfur chemistry of an extragalactic source, the nuclear region of the starburst galaxy NGC 253. This is the first time that H2S and, tentatively, H2CS are detected towards the nucleus of a starburst galaxy. Source averaged fractional abundances of these molecules are a few 10^-9, except for CS and OCS which are more abundant (10^-8). Sulfur isotopic ratios, 32S/34S~8+-2 and 34S/33S>9, are measured through observations of 13CS, C34S, and C33S. A comparison with the observed relative abundances towards different prototypical Galactic sources suggests t...