Ok, so let's say there is 350 km in an almost straight line. A great tunnel boring machine excavates around 20 meters per day in average (80 m on good days; next to nothing on other days). So that's 140 meters per week or 7 km per year. With 10 of those he could cover the distance in 5 years. For one tube. If he wants 2, 3 or 4 tubes, which would make sense on such busy stretch, he would need 2, 3 or 4 machines more. Musk claims the Boring Company will drill faster than traditional tunnel boring machines. If he could go twice as fast, he could do with half the machines.
It's doable, I guess, but I think the regulatory obstacles, archeological excavations, etc would slow it all down more than the machines.