Do you have well water or city water?
We had a well once, and the water had a sulfur smell to it, especially when hot. We called a water quality company and they told us the sulfur wasn’t harmful, just yucky. They were able to set up a water filter system that got rid of the smell.
For a simple fix, you can try this and see if it works. If the sulfur smell is caused by bacteria in your pipes, that may get rid of it. Ours was caused by minerals in the well water.
I would either call a water quality company to test your water and find the source of the smell or, if you are on city water, call the water company. I guess the smell can also be caused by a certain part on a hot water heater (which wouldn’t explain why you smell it outside) or even by a sewage problem.
Because I am paranoid, I’d use bottled water for drinking, cooking, and brushing teeth until the problem was solved.
edited to add: If you have a septic tank, check to see when it was last emptied. If you have a neighbor with a septic tank, that could be the source of the smell but it shouldn’t affect your water unless you have a well that is too close to the septic tank.