Where to find the date code
Look on the sidewall for a string starting with DOT, followed by letters and numbers. The last four digits are the date: the first two are the week, the last two are the year. 2321 means the 23rd week of 2021. The full code is often only on one side of the tire, so check the inner sidewall if you can't find it.
How old is too old?
Rubber hardens and cracks with age even when the tread looks new. Many car and tire makers recommend inspecting tires from about 6 years old and replacing them at 10 years no matter how much tread is left. On a car that's driven hard or tracked, old rubber has noticeably less grip well before it looks worn.
Old tires on low-mileage cars
Low-mileage enthusiast cars often sit on tires with great tread that are a decade old. Budget for a new set if the date code says so, and use it to negotiate. Sellers rarely mention it.