Is it worth buying a used iPod?

Often yes, for a click-wheel model in working order: it plays your own music files, with no apps and no ads. Apple stopped selling the iPod classic in 2014, so batteries and repairs now come from independent shops, and many used iPods are sold already modified.

Check before you pay

Tick them off as you go:

The model number engraved on the back (A1238 for the iPod classic) matches the listing.
It charges and plays, and the click wheel responds all the way around.
The battery: ask how long it plays on a full charge, or when it was replaced.
Modified iPods (new battery, flash storage instead of the hard drive) are fine when the listing says so clearly.
You can load your music: you need a computer that still syncs it (Finder on a Mac, iTunes on Windows).

Red flags

A hard-drive iPod that clicks, or freezes on the Apple logo.
"Untested" or "for parts" at a working price.
Stock photos instead of photos of the actual iPod.

Which ones still make sense

iPod classic (model A1238, sold 2007 to 2014): the most sought after.
Older click-wheel iPods and the iPod mini: popular, and often due for a new battery.
Later iPod nano and shuffle models: cheaper and simpler; no iPod shuffle has a screen.

Live listings on eBay

A few current listings, United States. Asking prices; read each listing's condition and returns.

Sources: Apple model numbers engraved on each iPod; Apple sold the iPod classic from 2007 to 2014.