Can you replace valve stem packing without shutting off the faucet?
This a common problem with old multi-turn valves. The first thing to try is to tighten the packing nut on the shut-off shaft. Sometimes this will compress the packing enough to stop the leak.

Trying to fix this packing while the water is on is very risky. Even with all the water outlets in the house turned on, you will still be seeing 50 to 70 PSI coming in from the street. You are going to see huge amounts of water coming out of the valve body, and if the new packing or cartridge doesn't go in quick and accurately, you have a potential disaster on your hands.  I would recommend that you call the water company, schedule a street turn off and replace the valve with a 1/4 turn stainless steel ball valve. The swap over only takes a few minutes, and the water company can either wait or come back later in the day to turn the water back on.