Usually what causes lots of hum is a shot electroytic filter capacitor. Usually 50uF at 160V, and another 33uF at 160V would work as replacements. The amount of uF's is not critical, but the voltage rating must exceed 150V. Places like Mouser.com or DigiKey.com will have suitable replacements (you need not worry about having both caps in the same package, cardboard tube or metal can).
You should also replace any and all wax paper caps.
Typical schematic of this sort of radio:
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