Language evolution is like biological evolution – it happens minutely, generation by generation, so there’s no distinct breaking point between one language and the next language that develops from it. Therefore, it’s impossible to say that one language is really older than any other one; they’re all as old as humanity itself. That said, each of the languages below has a little something special—something ancient—to differentiate it from the masses.
Hebrew is a funny case, since it essentially fell out of common usage around 400 CE and then remained preserved as a liturgical language for Jews across the world. However, along with the rise of Zionism in the 19th and 20th century, Hebrew went through a revival process to become the official language of Israel. While the modern version differs from the Biblical version, native speakers of Hebrew can fully comprehend what is written in the Old Testament and its connected texts. As the earliest speakers of Modern Hebrew often had Yiddish as their native language, Modern Hebrew has in many ways been influenced by this other Jewish language.
Tamil, a language spoken by about 78 million people and recognized as an official language of India, Sri Lanka, and Singapore, is the only classical language that has survived all the way through to the modern world. It is a member of the Dravidian language family, which includes a number of languages native mostly to southern and eastern India. Researchers have found inscriptions in Tamil dating back to the third century BCE, and it has been in continuous use ever since. Unlike Sanskrit, another ancient Indian language that fell out of common usage around 600 BCE and became mostly a liturgical language, Tamil has continued to develop and is now the 20th most commonly-spoken language in the world.
The language family that most European languages belong to is Indo-European, but they started splitting apart from each other probably around 3500 BCE. They developed into dozens of other languages like German, Italian, and English, gradually losing the features that they had all shared. One language, however, up in the Baltic language branch of the Indo-European family, retained more of the feature of what linguists call Proto-Indo-European (PIE), which is the language that they postulate was spoken around 3500 BCE. For whatever reason, Lithuanian has kept more of the sounds and grammar rules from PIE than any of its linguistic cousins, and can therefore be called one of the oldest languages in the world.
In case you haven’t heard of Farsi, it’s a language spoken in modern day Iran, Afghanistan, and Tajikistan, among other places. You’ve probably heard of Persian, and it probably conjures up pictures of genies coming out of bottles. They’re actually the same language, under a different name. Farsi is the direct descendant of Old Persian, which was the language of the Persian Empire. Modern Persian took form around 800 CE, and one of the things that differentiates it from many modern languages is that it has changed relatively little since then. Speakers of Persian today could pick up a piece of writing from 900 CE and read it with considerably less difficulty than an English speaker could read, say, Shakespeare.
Icelandic is another Indo-European language, this time from the North Germanic Branch (just for comparison, English is also a Germanic language, but from the West Germanic branch). Many Germanic languages have streamlined themselves and lost some of the features that other Indo-European languages have (you’ve probably never heard of a case, for example, unless you’ve studied Latin or a Slavic language), but Icelandic has developed much more conservatively and retained many of these features. Danish governance of the country from the 14th to the 20th century also had very little effect on the language, so it has mostly gone unchanged since Norse settlers brought it there when they came to the country, and Icelandic speakers can easily read the sagas written centuries ago.
The Slavic language family, which includes Russian, Polish, Czech, and Croatian, among others, is relatively young as far as languages go. They only started splitting off from their common ancestor, Common Slavic (or Proto-Slavic), when Cyril and Methodius standardized the language, creating what is now called Old Church Slavonic, and created an alphabet for it. They then took the language north with them in the 9th century as they went to convert the Slavs to Christianity. They came from somewhere just north of Greece, probably in what is now Macedonia, and Macedonian (together with its very close relative Bulgarian) is the language that is most closely related to Old Church Slavonic today.