Context. Stars form predominantly in groups usually denoted as clusters or associations. The observed stellar groups display a broad spectrum of masses, sizes, and other properties, so it is often assumed that there is no underlying structure in this diversity. Aims. Here we show that the assumption of an unstructured multitude of cluster or association types might be misleading. Current data compilations of clusters in the solar neighbourhood show correlations among cluster mass, size, age, maximum stellar mass, etc. In this first paper we take a closer look at the correlation of cluster mass and radius. Methods. We use literature data to explore relations in cluster and molecular core properties in the solar neighbourhood. Results. We show that for embedded clusters in the solar neighbourhood a clear correlation exists between cluster mass and half-mass radius of the form Mc = CRc γ with γ = 1.7 ± 0.2. This correlation holds for infrared K-band data, as well as for X-ray sources and clusters containing a hundred stars up to those consisting of a few tens of thousands of stars. The correlation is difficult to verify for clusters containing fewer than 30 stars owing to low-number statistics. Dense clumps of gas are the progenitors of the embedded clusters. We find almost the same slope for the mass-size relation of dense, massive clumps as for the embedded star clusters. This might point to a direct translation from gas to stellar mass: however, it is difficult to relate size measurements for clusters (stars) to those for gas profiles. Taking multiple paths for clump mass into cluster mass into account, we obtain an average star-formation efficiency of 18%-5.7 +9.3 for the embedded clusters in the solar neighbourhood. Conclusions. The derived mass-radius relation gives constraints for the theory of clustered star formation. Analytical models and simulations of clustered star formation have to reproduce this relation in order to be realistic.