Distance is 408 kilometers or 253 miles or 220 nautical miles
The distance is the theoretical air distance (great circle distance). Flying between the two locations' airports can be a different distance, depending on airport location and actual route chosen.
Map - Shortest path between Dallas and San Antonio
Location for Dallas
Location for San Antonio
The map is using a projection that makes land and oceans much wider near the south and north poles. The heading/course/bearing during a flight varies in most cases. Map based on image from NASA.