Unlike graphs, determining the Tur\'{a}n density of hypergraphs is known to be notoriously hard in general. The essential reason is that for many classical families of $r$-uniform hypergraphs $\mathcal{F}$, there are perhaps many near-extremal configurations with very different structure. Such a phenomenon is called not stable, and Liu and Mubayi gave the first example that is not stable. Another perhaps reason is that little is known about the set consisting of all possible Tur\'{a}n densities which has cardinality of the continuum. Let $t\ge 2$ be an integer. In this paper, we construct an finite family of $\mathcal{M}$ of 3-uniform hypergraphs such that the Tur\'{a}n density of $\mathcal{M}$ is irrational, and there are $t$ near-extremal $\mathcal{M}$-free configurations that are far from each other in edit-distance. This is the first example that has an irrational Tur\'{a}n density and is not stable. We also prove its Andr\'{a}sfai-Erd\H{o}s-S\'{o}s type stability theorem.