The translation given by Google Translate is, as typical with Latin, gibberish.
Neither the BBC translation nor the intended meaning get close to what the Latin says.
The quoted professor is being polite; it's a bit hard to translate nonsensical text.
I really wish people stopped relying on Google Translate in matters of any importance.
Trying to use similar words, I offer these translations:
- "Room in the house. Local people." — Locus in domo. Populus localis.1
- "Local homes for local people." — Domus/domos locales hominibus localibus.2
- "There should be some local homes for the local people." — Domos locales hominibus localibus offerre debemus.3
I would translate Sit ibi locorum domos ad loci populum. somewhere along these lines: "May there be houses of places to the people of the place."
But it is ungrammatical; with sint domus it would really mean what I interpret to mean, with sit domos I don't know how to parse it.
Mocking Google Translate again is left as an exercise.
As C. M. Weimer points out in the comments below, the adjective localis is not classical.
I still think consider it correct, but if you want something more classical, replace all occurrences of localis, locales, and localibus by huius loci ("of this place").
The word locus might not be the optimal choice, but I think it can be used this way.
The best choice of words depends on what is exactly meant: is there a room in the house, or is there room in the house?
And I'm not sure what the added "Local people." is supposed to signify…
The choice of case depends on intended nuance.
If it means "here are local homes", I would go with the nominative domus.
If "we offer local homes", then the accusative domos.
The Latin means more literally "we should offer local homes for local people".
This sounds more natural to me than a more direct translation.