We assume that when you say it won't start, you mean that the battery dies slowly over time and that within some relatively short time, it is dead.

This is likely caused by a slowly leaking circuit. There may be a short circuit that is not blowing a fuse, but slowly allowing electricity to flow, thereby draining the battery.

There may be a small -- almost invisible -- contact point where a live wire is in contact with the chassis or other grounded metal.

Check all of the radio's support frames for bare wires or even a braided wire with some few strands protruding, not insulated and touching the dashboard frame.

Also consider the radio itself. Is it a new or used radio? It could be defectively leaking electricity through the radio's light or other circuit.