Shutoff replacement procedure
If you get drips of water coming through your pipes even with the main shutoff valve completely closed, it's probably time to replace the shutoff with a new ball valve. It's not complicated, but the house water will be off, so it's a good idea to get your ducks in a row before you start. To remove the old valve, you'll have to undo the coupling on the “house side” of the water meter. There's usually an oil-impregnated leather sealing washer inside the coupling. Leather washers are tough to find, so look at a plumbing supplier or check for them online. Some plumbing suppliers sell neoprene washers, but the “oldtimers” swear by leather, because eventually neoprene will dry out and crack. You probably have a 3/4-in. pipe leading to and from the leaky valve. Buy a threaded replacement ball valve; it's a far better choice than the older gate-style valve.
Test the “street-side” valve where the water enters the house. Since that valve is just as old, test it to make sure it closes all the way and reopens again. If that valve needs replacement, contact your water utility to shut off the main valve at the curb.
Then check out the electrical system “bonding jumper” that runs from a clamp on the house side of the meter to a clamp on the street side. If you can unscrew the old valve with the bonding jumper wire in place, fine. If you can't, do not disconnect it. The safe way to work around this problem is to install a longer section of copper wire and two new clamps (6AWG for 100-amp service, 4AWG for 200-amp service). Then remove the short bonding jumper. At that point, you're ready to shut down the water and replace the valve as shown.