Acquiring high-spatial-resolution (HSR) images at high temporal sampling rate is not economical and even not achievable using contemporary multispectral satellite imaging hardware. An alternative is to fuse a set of HSR images acquired at low sampling rate, with another set of low-spatial-resolution images acquired at high sampling rate, and such fusion problem is referred to as spatiotemporal super-resolution (STSR). We mitigate the ill-posedness of the STSR problem by incorporating the image self-similarity prior (S2P), which is the key behind the design of several state-of-the-art imaging inverse problems. Unlike most super-resolution works in the computer vision area, our method does not rely on collecting big data. Instead, we propose a fully unsupervised STSR method by adopting the popular strategy in machine learning, known as plug-and-play optimization, and by carefully refining the required matrix computation/inversion. We term our method as STSRS2P, whose superiority and low computational complexity will be experimentally verified.