Wi-Fi fingerprinting-based indoor localization has received increased attention due to its proven accuracy and global availability. The common received-signal-strength-based (RSS) fingerprinting presents performance degradation due to well-known signal fluctuations, but more recently, the more stable channel state information (CSI) has gained popularity. In this paper, we present SDR-Fi, the first reported Wi-Fi software-defined radio (SDR) receiver for indoor positioning using CSI measurements as features for deep learning (DL) classification. The CSI measurements are obtained from a fast-prototyping LabVIEW-based 802.11n SDR receiver platform. SDR-Fi measures CSI data passively from pilot beacon frames from a single access point (AP) at almost 10 Hz rate. A feed-forward neural network and a 1D convolutional neural network are examined to estimate location accuracy in representative testing scenarios for an indoor cluttered laboratory area, and an adjacent, covered outdoor area. The proposed DL classification methods leverage CSI-based fingerprinting for low AP scenarios, as opposed to traditional RSS-based systems, which require many APs for reliable positioning. Demonstration results are threefold: (a) A fast-prototyping SDR platform that passively extracts CSI measurements from Wi-Fi beacon frames, providing a genuine possibility for vendor network cards to provide such measurements, (b) two state-of-the-art DL classification methods outperforming traditional RSS-based methods for low AP scenarios, (c) a testing methodology for performance evaluation of the proposed indoor positioning system.