How It All Goes Down
Ezra
As the curtain rises and the orchestra dramatically swells, King Cyrus of Persia has just defeated the Babylonians. Inspired by God's spirit, he tells the Israelites that they can head home and rebuild their temple. He returns the sacred temple vessels stolen by the Babylonians and personally bankrolls the whole building project from his treasury
Zerubbabel and Jeshua the High Priest (not Star Trek characters, but they should be) lead the people back and start laying down the foundations for a new Temple. But Israel's enemies are able to frustrate the building plans by getting the new Persian king Artaxerxes to order construction to a halt.
Things stay that way until Darius comes to power in Persia. Two prophets, Haggai and Zechariah, urge the Israelites to start building, so naturally the enemies of Israel again complain to the king and demand he check the records to see if Cyrus actually gave permission to build. But—bwahahaha!—when Darius finds Cyrus's original decree, he lets them start rebuilding. Ezra (earlier in time but not in the narrative) is sent by Artaxerxes to help the Israelites get their religious observance back on track. A big part of this involves Ezra breaking up marriages between Israelites and non-Israelites. He has a major meltdown when he hears about all the intermarriage, and manages to convince the men to send away their foreign wives and children.
Nehemiah
Nehemiah's a Hebrew cup-bearer to Artaxerxes, who allows him to return to Jerusalem to help set things in order and rebuild the city. Gentile leaders from different provinces (specifically, leaders of the Ammonites, Samaritans, and Arabs) try to derail the rebuilding project, but thanks to some military readiness and cases of Five-Hour Energy, Nehemiah and his workers successfully speed-build the walls of the city.
After this, Nehemiah reads the riot act to Israelite officials and nobles who have been oppressing the poor, charging ridiculous interest on loans, and forcing the people to pawn their land in order to eat. He successfully evades charges of rebellion against the Persian king drummed up by his enemies, and has Ezra give everyone a crash course on the Laws of Moses. The surviving Jews return from exile to repopulate Jerusalem, and Nehemiah—who's come back from a trip to the Persian capital in Susa—shapes up the backsliding Jews, breaks up more interfaith marriages, and saves the day. And he's the first to admit it.