There are many small,old houses across the USA. You can search via Zillow for houses with sq ft between 100 and 600 (or whatever numbers you choose).
I usually start with Zillow for its easy search features, then move to Realtor.com because it’s more accurate.
Here’s a cute little house in Cleveland, 441 sq ft for only $28,900, including the land. It would be hard to build a new tiny house for much less, even without land.
Be sure to have the home inspected. Older houses may have termites or mold and/or need new wiring.
If you (or the house) won’t qualify for a traditional mortgage, you may be able to get a loan through private lenders or hard money lenders.
Here’s the listing for the house pictured http://www.realtor.com/realestateandhomes-detail/13421-Sprecher-Ave_Cleveland_OH_44135_M39269-20012