That depends on the cabbie. For the right price, they will take you cross country.
It s not illegal for the taxi to take an adult between states, but it is illegal for them to transport a minor who may be running away from home or otherwise traveling without parental permission. That could be a federal crime which would land the taxi driver in jail.
Of course you can. You can walk, drive, take a bus or a taxi. There are no border guards at state borders.