How much can I sell my iPhone 4 for?

I've had my iPhone 4 for 1 year and 8 months. I bought it refurbished. It has 3 problems:

-The back is cracked pretty badly. The screen is in perfect condition, though.
-It recently started doing this thing where, no matter how much battery I have, it will turn off and won't turn back on unless I plug it in.
-Yesterday the volume up button stopped working, so I have to go into settings if I want to turn the volume up.

I'm tired of all these problems and want to get a new phone, so maybe someone would be desperate enough to buy it from me. How much do you think it is worth?

Not much, if damaged then value goes way down. Plus that hardware came out about 4 years ago or more.

$5.00.

It's an obsolete model in need of a new back and likely a new battery and it needs time to diagnose the causes of the other problems and then more parts to fix it.

That's going to cost more than the phone is now worth so no retailer will be interested in it except as a possible parts donor, but the only part likely to be of use is the screen. So it's worth a little less than a retailer would pay for a new screen as yours still needs to be removed from your scrap phone and they then have to pay to dispose of the remaining hazardous waste.

But to a private individual who wants it to practice their repair skills on it might be worth about £5.

Ebay is full of idiots with more sense than money though so advertise it there with a full and honest description of the faults and you might find someone who'll pay more.