We introduce a novel received signal strength intensity (RSSI)-based positioning method using fluid antenna systems (FAS), leveraging their inherent channel correlation properties to improve location accuracy. By enabling a single antenna to sample multiple spatial positions, FAS exhibits high correlation between its ports. We integrate this high inter-port correlation with a logarithmic path loss model to mitigate the impact of fast fading on RSSI signals, and derive a simplified multipoint positioning model based on the established relationship between channel correlation and RSSI signal correlation. A maximum likelihood estimator (MLE) is then developed, for which we provide a closed-form solution. Results demonstrate that our approach outperforms both traditional least squares (LS) methods and single-antenna systems, achieving accuracy comparable to conventional multi-antenna positioning. Furthermore, we analyze the impact of different antenna structures on positioning performance, offering practical guidance for FAS antenna design.