Specifically, what is the oldest language that we know how to speak (including exact pronunciation, voweling, slang, etc.) well enough to be confident that we could converse orally with an ancient native speaker? And how do we know what these letters and words sounded like if there was nobody speaking it at time of (re)discovery?
I don't know what happened to the previous answer, since it made good points, but I'll give you a new one.
Languages evolve. They change over time, and this is a gradual process, that is often hard to pin down in terms of mutual intelligibility between different eras. Of course, this is rarely a problem, since people from different eras never meet (although cryogenics might change that in the future) and have no need to communicate orally. Writing is a different case, as it spans longer time frames, but that is not your question and could probably fill another thread.
To find out, how a language was pronounced, I cite William Allen Vox Latina—a Guide to the Pronunciation of Classical Latin, where on p. viii of the introduction, he explains what resources he used.
(1) specific statements of Latin grammarians and other authors regarding the pronunciation of the language; (2) puns, plays on words, ancient etymologies, and imitations of natural sounds; (3) the representation of Latin words in other languages; (4) developments in the Romance languages; (5) the spelling conventions of Latin, and particularly scribal or epigraphic variations; and (6) the internal structure of the Latin language itself, including its metrical patterns.
By analyzing these and rooting out a few contradictions, you are able to paint a coherent picture of how a language was spoken during a specific period of time.
We linguists take these languages at the oldest states possible, and from there reconstruct a proto-language, that is the ancestor of a certain group or family of languages. So for example by comparing Latin, Old Greek, Gothic, Sanskrit and so on, you can reconstruct the ancestor of all Indo-European languages, Proto-Indo-European (PIE). To a certain degree you can do the same with Latin (he does in (4) above), because it is practically Proto-Romance. Reconstructing - to a certain degree - a proto-language is how you prove beyond a reasonable doubt, that a selection of languages belongs to one language family.
For unattested Proto-language, almost all of the options above are unavailable, only No 4 remains. But PIE was a very hot topic in the early stages of historical and comparative linguistics, so that today you could write a text in PIE reasonably well. Common expressions like How do you do? are harder to reconstruct, since their use changes very fast and has only limited connection to the semantics of its constituents (You don't need to do anything in a specific way (how) to feel fine.) So every day conversations and small talk would be hindered.
Another thing to remember about reconstructed languages like PIE is, that there is no archeological evidence for them, that would allow us to place them chronologically or geographically. Through secondary evidence, we can place for example Proto-Bantu in Western Africa, despite Bantu languages today being mostly spoken in Eastern and Southern Africa. But this is not based on findings of the language, but rather all the other stuff showing migration (farming implements etc.). This is why the time estimates given to when these Proto-languages are spoken, vary widely, if not more. PIE is estimated around 4000-2000 BC, so is Proto-Bantu. Proto-Uralic might have been spoken somewhere between 7000-2000 BC. To pick one as being the oldest is therefore practically impossible.
Regarding your question about the accuracy of these reconstruction: The more languages you have to recreate a proto-language from, and the older these languages are, the better is the reconstruction, presuming it succeeds. Since we have well attested languages from centuries ago from various Indo-European language groups, I would rate the reconstructed PIE as very reliable. For the Japonic languages, the sample is much smaller, but the split into Japanese and Ryukyuan language is much more recent (Yamato period, 250 - 710 AD), and Old Japanese (Heian period, ca. 800 - 1200 AD) is very well documented. It is therefore easier to reconstruct Proto-Japonic, since you just need to go back a few centuries.