Bananas throw out beta radiation. It's not as penetrating as alpha - attenuation depth in water on the order of centimeters rather than millimeters? - but I think a banana-equivalent dose will be pretty solidly concentrated in your digestive tract.
However, you also excrete potassium making the Banana equivalent meaningless as it's calculated by assuming you kept that potassium for 50 years which simply does not happen. AKA you would end up with less than 365 Banana equivalent doses by eating 1 banana a day for 1 year. (https://en.wikipedia.org/wiki/Banana_equivalent_dose)
Humans are surprisingly radioactive BTW. We each have about a hundred grams of potassium happily radiating away. An interesting technological challenge for a startup might be using purely passive radiation monitoring to detect humans. For security sensing or something. Maybe in a 100 years automatic supermarket style doors will sense humans radioactively rather than current microwave doppler and infrared systems. Maybe.
Beta is super-easily stopped. A piece of cardboard or tinfoil reduces it dramatically, or stops it completely, depending on energy per particle, total flux, etc.
I've a Luminox watch, the dial is visible in darkness all the time, you don't have to "recharge" it. It works by having small amounts of tritium decay within luminescent tubes. The radiation produced is beta - but the levels of radiation outside the watch are essentially zero. The body of the watch is enough to contain it.
That's the wrong way around. Alpha radiation barely penetrates at all.