This paper studies a stochastic congested location problem in the network of a service system that consists of facilities to be established in a finite number of candidate locations. Population zones allocated to each open service facility together creates a stream of demand that follows a Poisson process and may cause congestion at the facility. The service time at each facility is stochastic and depends on the service capacity and follows a general distribution that can differ for each facility. The service capacity is selected from a given (bounded or unbounded) interval. The objective of our problem is to optimize a balanced performance measure that compromises between facility-induced and customer-related costs. Service times are represented by a flexible location-scale stochastic model. The problem is formulated using quadratic conic optimization. Valid inequalities and a cut-generation procedure are developed to increase computational efficiency. A comprehensive numerical study is carried out to show the efficiency and effectiveness of the solution procedure. Moreover, our numerical experiments using real data of a preventive healthcare system in Toronto show that the optimal service network configuration is highly sensitive to the service-time distribution. Our method for convexifying the waiting-time formulas of M/G/1 queues is general and extends the existing convexity results in queueing theory such that they can be used in optimization problems where the service rates are continuous.