350 miles. This is the route:
Dallas to Pensacola is 650 miles. Dallas to Key West is 1470 miles.
The distance in Florida from Pensacola to West Palm Beach is 506 miles. That equals 815 kilometers or 440 nautical miles.
Pensacola is 1866 miles from Las Vegas. Key West is 2686 miles.
Springfield to Pensacola is about 730 miles. Kansas City to Key West is about 1630 miles.
The technical term would be the "antipode". For Pensacola Florida, this is a spot in the Indian ocean about 2000 miles west of Perth, Australia. The nearest dry land is probably the Kerguelen Islands, and even those are well over a thousand miles away.
The address of the West Florida Public Library is: 200 West Gregory Street, Pensacola, 32502 4822
Pensacola to Monterrey is just over 1000 miles. Key West to Cancun is just over 3100 miles.
The address of the West Florida Genealogy Library is: 5740 North Ninth Avenue, Pensacola, 32504 8970
The address of the West Florida Genealogical Society Inc is: Po Boxx 947, Pensacola, FL 32591-0947
Google Maps says 358 miles from Jacksonville, FL to Pensacola, FL. Add another 10 miles or so to get to the Alabama line, and you're around 370 road miles. Google Earth shows it as 360 miles "as the crow flies", so those numbers agree nicely.
north west
If you want to find the shortest distance from the east coast to the west coast then Florida is considered an east coast state. It's only 1763 miles from the coast at Pensacola, Florida to San Diego.