Do you mean the stereotypical hostage-holdout situation, with multiple random people inside and the building surrounded? Or do you mean any hostage-taking at all?

Because if it's the latter the answer is probably going to be whatever the oldest war is that we have good records for. Prior to the modern age hostage-taking was politics 101. If you wanted a vassal to be loyal, you took hostages. If you wanted to secure an agreement, you exchanged hostages.

The earliest specific incident I can point to is the Third Punic War, 149 BC when the Romans demanded 300 noble Carthaginians hostages as terms of peace (they didn't honour it, but that's neither here nor there). You get the feeling from records of the city's reaction that this is an unusually large number, but it's clear that by this time hostage-taking is well-established tradition.

I'm also certain that my more learned colleagues can provide many, many earlier instances.

In Chinese history 公羊传:

庄公十三年 ... 冬，公会齐侯，盟于柯。... 庄公升坛，曹子手剑而从之。管子进曰：「君何求乎？」曹子曰：「城坏压竟，君不图与？」管子曰：「然则君将何求？」曹子曰：「愿请汶阳之田。」管子顾曰：「君许诺。」桓公曰：「诺。」曹子请盟，桓公下与之盟。已盟，曹子摽剑而去之。

Rough translation: year 680 BCE winter, Duke of Lu met Duke of Qi at Ke for peace treaty. At the meeting, Duke of Lu's general Cao suddenly drew sword and got behind Duke of Qi. Qi's chancellor Guan Zhong asked: "What do you want?" Cao said: "You took so much of our land, what do you think? I'd ask you to return (certain area) to us. " Guan Zhong looked at Duke of Qi and asked: "My lord please promise so. " And Duke of Qi promised, then they had the pledging ceremony. Then Cao threw down his sword and walked away like a boss. (Later Guan Zhong persuaded Duke of Qi to honor his word)

You can say this is a kind of hostage situation.

Another piece, in 三国志， it is recorded that Cao Cao (probably not related to the General Cao above) in year 192 CE made it into law that officials are NOT to be responsible to the safety of hostages and should always attack the hostage-taker, this law greatly reduced the number of hostage taking incidents; this implies hostage situation was a common thing in the chaotic years of late Han dynasty before 192 CE.

Its been recorded that most of the population of Jerusalem were held hostage by Jewish fanatics during the cities siege in the Jewish revolt at the end of Nero's reign. Josephus and the Talmud both claim that the majority of Jews in the city were opposed to continuing the war and wished to negotiate with the Romans. As it was in Vespasian's the Roman commander, ( although the actual siege was overseen by his son Titus.) interest to quickly end the revolt. negotiations may have been possible. However the minority group of fanatics were said to prevent this by using terror tactics on their own people. The result being mass starvation, reports of cannibalism, destruction of the Temple, slaughter of vast amounts of people, with the enslavement of the survivors. Many of which ended up in the Arena. Not the Coliseum. It had not yet been built. Of course this should be taken with a grain of salt. Its possible they were writing to appease the Romans. Certainly in Josephus's case its very likely. He basically defected to the Romans to save his neck.

It would also depend on your definition of "Hostage". It was common in almost all Ancient society to send important people, usually royal family children to seal a treaty. As a rule these were treated well and educated but they were under no delusion of their fate if the treaty was broken. An example of this practice was Alexander the Greats father Phillip. He was sent to Thebes spending much of his childhood there. In Japan the future Shogun Tokugawa was also a "Hostage" during his Youth.