The home inspection contingency is to look for defects in a home. A defect is defined as something that can significantly threaten the safety of the home's occupants, the value of the home or the life of the home. Key word "significantly".
Only an attorney can tell you if what you stated was a defect, but I tips towards no. Both are working properly and apparently cause no threat as stated above. These would not be "walk away" items.
That said, you could ask for anything in your home inspection contingency, even if that were to pain the entire house pink. It is up to the seller to decide to do it for you. Were I the seller's agent, I would advise them that the equipment may be old but it is working as expected. They don't "need" to fix it to keep you in the deal. They likely won't fit the definition of defect either, so these items would probably not be firm grounds for canceling the deal.
My best advice to you is if the house came out great on the home inspection stay in the deal and budget for a new furnace and central air. If you maintain the existing units well, they may last a long, long time. The phrase your home inspector probably used is "at the end of their EXPECTED life". These things don't just croak after a certain amount of time. Maintain it and you could get many more years out of it.
Best of luck.
You are smart to have inspections done. In a multiple offer situation (and particularly if there is a back up), the seller may not be inclined acquiesce to requests for repairs, if any.
You can ask, but he can say no. If you back out, he will put it back on the market. EXCEPT ---- your inspections now become part of the disclosure package, and other buyers may raise the same questions.
Just for reassurance....and something you may ask for...get a home warranty to cover these items. They may cost extra, but it's worth your peace of mind. And renew it for as long as you want.
Good luck, and congratulations on getting your offer accepted in a multiple offer situation.
I agree with Karen...see if you can negotiate a home warranty to be included with the purchase price. Alternatively, you as the buyer can also purchase a home warranty. That will give you peace of mind for at least the first year...this is typically when new buyers are stretched financially and can't afford to replace major mechanicals. You will also want to continue to get the furnace and A/C serviced on a regular basis...no one knows when the furnace and A/C will "croak," but keep in mind that someday it will, so you will just need to budget for a future replacement. Much like the brakes and tires on your car...at some point they will need to be replaced!
Have these conversations with your agent and I'm sure he/she can guide you through this process!
I'm sure you will find a good resolution. Enjoy your new home!