MacBook Air turned off unexpectedly and won’t power or charge back on

...I was playing a game on it, and suddenly after 4 hours...

😳

That's a huge clue. Mac notebooks, regardless of model, have limited space for fans and the airflow need to defend against high-load functions like gaming. You problem is likely that most Macbook Airs have NO cooling fans. Do a web search for "does Macbook Air have a cooling fan" and look a the results.

Gaming on a fanless Macbook Air is equivalent to driving your car down the motorway at high speed with the cooling system removed.

Have you tried restarting since the computer cooled down? If that does not work, you've probably fried your computer. Hopefully Apple will honor its no-questions-asked return policy.

HOWEVER, that policy ONLY applies if you bought directly from Apple through their online site or an Apple-owned retail store. If bought from an Apple Authorized Reseller or a non-Apple third-party, Apple states that the SELLER's return policy applies.

Returns & Refunds - Shopping Help - Apple

If you are able to return your recently-deceased computer for a refund, buy something more suitable for gaming. Not a laptop.

IF you purchased directly from Apple, return it. They will allow returns for 2 weeks for ANY reason. Of course, I would do this AFTER calling AppleCare assuming AppleCare was not able to help you out.

AppleCare Contact Info