Since the 3rd gen Nano was released in 2007, I'm guessing it wasn't new, but it's always possible it was some unsold stock. Either way, the battery by now will have long outlived its useful life, so that's where I'd start; by replacing the battery.
Although you'd think it should at least run when it's plugged in, I've found that there are a lot of battery-powered devices that won't run unless the battery is capable of powering the device, whether it's plugged in or not.
Unfortunately for us home repairers, replacing the battery on this Nano requires soldering three wires. If you're comfortable doing that, then iFixit has a guide showing you step by step how to put in a new battery.
iPod Nano 3rd Generation Battery Replacement - iFixit Repair Guide
If you'd like to support iFixit's repair efforts, you can buy the battery right here on this site, but you can also find it easily on the major retailers like eBay, Amazon and AliExpress.
iPod nano (3rd Gen) Battery
If soldering is outside your comfort zone, you'll need to find a local repair shop that can handle the job; it's not complicated so it shouldn't cost too much but check around and get some quotes.
Good luck; let us know how it turns out!
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