We develop a tractable equilibrium model of stock-flow matching in the labor market. The economy consists of many labor markets, and workers and jobs continually flow into each labor market. Within a labor market, potential worker-job matches differ in quality. Accordingly, a worker newly arrived in a labor market may not find any acceptable matches; if so, she becomes part of the stock of unemployed workers and must wait for the arrival of a new vacancy which offers her a sufficiently high quality match. When labor market conditions change, the set of acceptable matches changes in response, a feature not allowed for in previous work on stock-flow matching. Our model is consistent with several stylized facts about the labor market, such as the importance of flows, as well as stocks, for matching rates, as well as with duration dependence in unemployment. While the model is stylized, it can account quite well not only for the shape of the US unemployment duration distribution, but for how it changes between high- and low-unemployment periods. Changes in the set of acceptable matches, the key innovation of our modeling approach, are important in allowing the model to do this.