99.99% of the time, when we use the word 'people', it is a plural noun. 'People' means men, women and children, in the plural. We tend to use as the plural form of 'person.' If you came across an situation in which a native speaker wrote 'that people', there are two explanations: 1. He or she could be using the word 'people', in the singular, to refer to a particular nation or ethnic group. For example, you can say 'The Aztecs were an ancient people who lived in Mexico'. In this sense, 'people' is singular. 2. It could be a mistake or typographical error.

You could say... Talking is something THAT PEOPLE do

Ozzie's inspired answer has given me another idea. Here's a third possibility: 3. The word 'that' isn't a demonstrative adjective at all. For example: 'I hope that people will buy this product'. In this sentence, the subordinate clause is simply 'people', and the 'that' is there to connect the two clauses. [I hope] [that] [people will buy this product].

No its not correct. you'd have to say that person, the singular of people, for it to make sense.