While anything is possible, and I have heard of people claiming to be sensitive to natural gas itself, what typically causes problems are the products of combustion of natural gas. That matches with "the exhaust came on". So the rest of this answer is based on the problem NOT being a natural gas leak (which would be often under control or at least supervision of the local natural gas utility) but rather a carbon monoxide or other combustion problem.
When a natural gas appliance (furnace, cooktop, water heater, generator, etc.) is running, the primary combustion products are water and carbon dioxide. However, if natural gas (or other carbon fuels) does not burn properly, due to insufficient oxygen or equipment malfunction, carbon monoxide (CO) can be produced. Carbon dioxide is harmless in small amounts but in large amounts can cause problems. Carbon monoxide is harmful even at low levels and can be fatal. CO is odorless and colorless and symptoms of CO poisoning are similar to the flu and often, tragically, not spotted in time. (And recently used as a way to nearly kill the main characters in a science fiction novel, https://www.amazon.com/Delta-v-Daniel-Suarez/dp/1524742414 , but I digress...)
The detection and resolution of a carbon monoxide problem, unlike a natural gas leak, is typically the responsibility of the homeowner and not the utility. Which means calling in a qualified HVAC technician to check the equipment and repair, replace or redesign portions of it as needed. The most important thing is to make sure that under normal operation, there is no significant amount of carbon monoxide either inside (e.g., near the furnace) or outside (the exhaust pipes). Assuming that is resolved, then there are two additional things to do:
Consider extending/redirecting the exhaust pipe for carbon dioxide. It is possible that if you are outside near the exhaust and it is running properly that the carbon dioxide could affect you. It shouldn't be a big problem, but if you have ruled out carbon monoxide then that is a possibility and the exhaust should be directed farther away. Typically it will be directed up as well - e.g., through the roof, though that may not always be an option.
Add carbon monoxide detectors. You should have one one on each level, and one not too far from any natural gas appliances. This is important because a combination of incomplete combustion combined with a cracked heat exchanger could result in carbon monoxide inside your house.