The US Census department provides a fair amount of data on rents within areas. Median Gross Rent, Average Gross Rent, you've got the normally bands of rents ($800-$999, $1000-$1249, etc). But these rent numbers could be for 1BR apartments or massive homes. Are there any columns that could shed some light on rents with respect to the size of housing units? I don't think the census department collects square footage of homes, but maybe bedrooms?
You want table B25068 'Bedrooms by gross rent' from the American Community Survey 5 year estimates for the most recent years (as of April 2017, that would be 2011-2015).
The ACS data has some pitfalls - pay attention to the margin of error because it'll show values for small geographies but they'll have margins that make them useless. If that comes as news to you then Google for a tutorial or post another question for more details.
For use in news reporting I was able to get to zip code or CDP subdivision level with reasonable margins of error. YMMV depending on use case. HTH!