It's an idiom that means "to bury someone".
Bodies are buried six feet below the surface. "Why are they?" you might ask.
Well when England was being ravaged by the Bubonic Plague in order to limit the outbreak, and spread of the disease, The Lord Mayor of London enacted a series of rules with regard to dealing with the bodies of those who died from the plague. This included a mandate that all dead must be buried in graves that had been dug a minimum of six feet deep. (Scientifically, this “solution” to bury bodies six feet under was ineffective in reducing the transfer of the plague since the fleas quickly leave its dead host and find a new live one)
Ok, so when the word foot is used as an adjective it should not be pluralised as there are no plural adjectives in English. For example: a match box, in that sentence the word 'match' is an adjective describing the box. So we would say "bury me six foot down" because the measurement (noun), foot, becomes adjectivial and describes the depth. However, in modern times when talking about measurement foot and feet have become interchangeable. However, when talking about distance you always say feet unless it 1 foot away.