Non-orthogonal multiple access (NOMA) has emerged as a new multiple access paradigm for wireless networks. Although many results have been produced for NOMA, most of them are limited to theoretical exploration and performance analysis in cellular networks. Very limited progress has been made so far in the design of practical NOMA schemes for wireless local area networks (WLANs). In this paper, we propose a practical downlink NOMA scheme for WLANs and evaluate its performance in real-world wireless environments. Our NOMA scheme has three key components: precoder design, user grouping, and successive interference cancellation (SIC). On the transmitter side, we first formulate the precoding design problem as an optimization problem and then devise an efficient algorithm to construct precoders for downlink NOMA transmissions. We further propose a lightweight user grouping algorithm to ensure the success of SIC at the receivers. On the receiver side, we propose a new SIC method to decode the desired signal in the presence of strong interference. In contrast to existing SIC methods, our SIC method does not require channel estimation to decode the signals, thereby improving its resilience to interference. We have built a prototype of the proposed NOMA scheme on a wireless testbed. Experimental results show that, compared to orthogonal multiple access (OMA), the proposed NOMA scheme can significantly improve the weak user's date rate (93% on average) and considerably improve WLAN's weighted sum rate (36% on average).