Although the thoughts on this aren’t completely settled, it’s likely Malachi was written in the middle of the 5th century BC. Since we last met Israel/Judah in the time of Amos, Israel was exiled by the Assyrians in the 8th century BC. Although Judah was ransacked, they weren’t exiled. Subsequently, though, Judah is attacked by Babylon in the late 7th century BC and fails to keep their agreement to pay Babylon to keep them at bay. Ultimately, Babylon attacks and exiles Judah in the early 6th century BC (between 597 and 560). Eventually, the exiles from Judah (and probably some of the remnants of the tribes of Israel) are allowed to return in 537 BC, although they are reduced people coming home (not all of them decide to come back) to a smaller amount of land.
This time is the start of Second Temple Judaism, in case you’ve heard that phrase. It indicates the use of a second temple after the first one , built by Solomon, was destroyed by the Babylonians. This temple is the one that will be in use between 517 BC all the way through 70 AD.
If you’ve read any of the prophets that were talking during the exile periods, coming back home and rebuilding the temple seemed to be a time when God’s people would see prosperity and blessing and the return of God’s presence (in the rebuilt temple, yes, but as a force in their lives in general also.) However, that doesn’t really appear to be the case. In fact, the argument for this kind of under-girds the premise of Malachi, which presents a fake conversation between God and the Jewish people, who are not satisfied with their lives after returning home. It’s interesting, the exiled Jews in Babylon weren’t treated particularly poorly and most chose to stay there rather than return to Judah. In fact, Judah is the smallest contingent of Jews compared to those who stayed in Babylon (conquered by Cyrus of Persia, the dude who let them go back) and those in Samaria who were left behind from the exile and who ultimately build a temple on Mount Gerizim and claim that is the true place God said for it to be constructed.