Contemporary Amperex Technology Co., Limited (CATL), the largest battery manufacturer in the world with a 38% share of the global market, has just announced some fairly significant breakthroughs in battery tech that aren't just theoretical; they're already hitting the market.
Why would it be impossible? A typical distribution power line is something in the range of 3MW, and that’s just the regular power lines running along the side of the road in your typical city. Larger power lines (i.e. long distance lines) have orders of magnitude higher voltage and, I assume, total power delivery.
So something official like a large recharge station (i.e. converted gas station) could pretty reasonably support 1-2MW if needed, though they’d probably end up w/ local storage so they don’t spike power demand like crazy, and then recharge that storage w/ solar in the day and base load at night.