If you mean words we have written evidence of, you would have to go back to the earliest writing systems. In Europe, the oldest deciphered system is Linear B, an old syllabary used in Mycenaean Greece (15th century BCE) to write their language (Greek). Linear B is based on an earlier writing system, Linear A, but that is undeciphered and it is unknown what language it was used for.

Linear B scripts have plenty of words that still exist, such as the word for king: basileus. Linear B was an awkward writing system which spelled this word with four characters representing the sounds "qa-si-re-u". This word is still used in English in loanwords from Greek such as "basilica" and "basilisk".

Words can be reconstructed from languages that were not written down. This is done by the comparitive method, which compares words in descendant languages and accounts for regular sound changes to arrive back at the word's original form with (in many cases) a great degree of confidence. The vast majority of English words can be reconstructed back to proto-Indo-European, spoken about 5000 years ago. Tons of words can be traced back this far. The word "word", for example, can be reconstructed as *were- from the proto-Indo-European word meaning "speak."

Other language families were spoken even farther back, such as Afro-Asiatic, spoken 10,000 years ago. But they are harder to reconstruct for that reason. But proto-Afro-Asiatic contributed a few words to English, and gave rise to modern languages such as Arabic and Hebrew.