Is it possible to buy a battery for a 5c apple iphone?

If you're talking from Apple, they don't provide OEM batteries for DIY installation in iPhone 5 models. And they've discontinued replacement service for the iPhone 5c.

As for an aftermarket battery - that's probably your only way at this point. I won't provide any links to the procedure or where to get one, but it does appear to be something I'd be OK with doing myself if I had the right tools.

I'll just say that one of the more reputable DIY outfits in the US charges less than $30 for a complete kit including the battery, that includes the tools. At least with a 5c there are no adhesives, which can be very difficult to deal with - even the adhesive seals in newer iPhones.

Even if it is, it's not worth it. A battery replacement would cost more that that device is worth. it won't even work as a phone in most of the world any more.

Not sure, Apple lists iPhone 5 models, but not the 5C. It would be best to contact Apple and ask them.

iPhone Battery Repair & Replacement - Apple Support