Sequential estimation is a well recognized approach to inference in statistical theory. In sequential estimation the sample size to use is not specified at the start of the study, and instead study outcomes are used to evaluate a predefined stopping rule, if sampling should continue or stop. In this article we develop a general theory for sequential estimation procedure for constructing a narrow confidence interval for a general class of effect sizes with a specified level of confidence (e.g., 95%) and a specified upper bound on the confidence interval width. Our method does not require prespecified, yet usually unknowable, population values of certain parameters for certain types of distributions, thus offering advantages compared to commonly used approaches to sample size planning. Importantly, we make our developments in a distribution-free environment and thus do not make untenable assumptions about the population from which observations are sampled. Our work is thus very general, timely due to the interest in effect sizes, and has wide applicability in the context of estimation of a general class of effect sizes.