How can I fix a leaky tap (exterior of the house) that I can't turn off the water source?
Turn off your house main, while fixing this.  It would behoove you to plan this when no other household members need water. And prepare to do it quickly.

I recently had a similar problem, but my house shutoffs didn't work either.
I had to call a plumber, who called the town, and shut off the water to my house at the street, while the plumber replaced both main shutoffs (either side of the meter) and replaced the leaky exterior faucet shutoff as well.  Best practices these days for shutoff valves are "ball valves" as they are less likely to fail than the traditional rubber washers on seats.