I did some anolysis regarding the map size https://imgur.com/l3fS5G4 For those too lazy to click that link, I came to the conclusion that the Fallout 4 map size is 9 square miles.
How did I arrive at that conclusion? I was looking at data I retrieved using the Companion App API. Fallout sends some 'calibration' coordinates (the edges of the red squared area in the image above) to the app, so that the app knows where to show the map markers on the global map. The coordinates the game sends are NW (-135168, 102400), NE (114688, 102400) and SW (-135168, -147456). With each local map snapshot the game sends to the app, it again sends some 'calibration' coordinates which can be used to determine where in the global world the local snapshot is placed. Example coordinates are NW (-42959.922, 59482.934), NE (-39089.203, 59482.934) and SW (-42959.922, 57305.652). They change all the time as you move around, but the width and height always stay the same. Comparing the width and height of global map and local snapshot we can learn that a local snapshot fits 64.55 x 114.756 times into the global map.
Next I needed some feature from the local map whose size I could estimate. I choose to use one of the big wood prefab tiles from the settlement workshop. Measuring the size on the local map revealed that you can fit such a tile 15.234 x 8.571 times into the local map. After some more calculations we know that a prefab tile fits 983.355 x 983.573 (there seem to be some minor measurement errors here as I suppose those two numbers should exactly be the same) times into the global map.
Assuming that the size of a prefab tile is 3x3 meters I arrive at the conclusion that the Fallout 4 map size is about 3000 x 3000 meters, or 9 square kilometers.