Human language is composed of sequences of reusable elements. The origins of the sequential structure of language is a hotly debated topic in evolutionary linguistics. In this paper, we show that sets of sequences with language-like statistical properties can emerge from a process of cultural evolution under pressure from chunk-based memory constraints. We employ a novel experimental task that is non-linguistic and non-communicative in nature, in which participants are trained on and later asked to recall a set of sequences one-by-one. Recalled sequences from one participant become training data for the next participant. In this way, we simulate cultural evolution in the laboratory. Our results show a cumulative increase in structure, and by comparing this structure to data from existing linguistic corpora, we demonstrate a close parallel between the sets of sequences that emerge in our experiment and those seen in natural language.