Additional links with press coverage: ArcheologyMag, Oxford.
For context:
The Huns were nomadic people from Central Eurasia; known for displacing a bunch of Iranian (e.g. Alans) and and Germanic (e.g. Goths, Suebians etc.) speakers, that ultimately invaded the Roman Empire. They reached the Volga around 370 CE, and one of their leaders (Attila) is specially famous. Often believed to be a Turkic people, but if the study is correct they're from a completely different language family instead.
The Xiōng-Nú are mentioned by Chinese sources as one of the "Five Barbarians" (i.e. non-Han people). They would've lived in Central Eurasia between 300 BCE and 100 CE or so, and eventually became Han tributaries.
The Paleo-Siberian language in question would be an older form of Arin, a Yeniseian language. Yup, that same family believed by some to have relatives in the Americas.