Eating, drinking and sleeping at this location is prohibited
might feel acceptable because:
- the structure is imitating what would be found in the case of participles being reduced into a nounphrase by conjunction reduction
- "at this location" restrictively and independently modifies three items in a list, which feels quite strange
If, for example, we take the sentence "the houses, shops and banks at this location are old", there's clearly no question of whether we should conjugate for the singular or plural, because there's no possibility of mechanism №1 coming into play, which just leaves us with the slightly awkward feeling conjured by having a modifying phrase that is implied three times but reduced to being written only once because we can do that in English to avoid repetition. In the case of the sentence with participles, because there is an akward feeling and because a way to escape it is provided by the existence of a different structure that is syntactically similar, speakers might naturally use the wrong construction (where the list of participles is reduced into one whole nounphrase (which we can identify by the singular subject-agreement)) just because it feels natural.
This is what I believe is going on here.