And even in those few days it probably won't affect things that aren't close to the accident.
A sodium 24 atom decays to an excited magnesium 24 atom by emitting a 1.39 MeV electron. That has a penetrating power of about 5 m in air.
The excited magnesium 24 quickly drops to the ground state by emitting two gamma rays, one at 2.76 MeV and one at 1.38 MeV. Those won't get past maybe 150 m of air.
In general, radiation emitted from a radioactive substance is not itself radioactive, nor is it particularly dangerous once it stops moving. The usual radiation consists of:
Alpha particles: literally just Helium minus the electrons. Only harmful because it moves fast at first. Even then, it’s only really harmful if it gets emitted inside your body. It barely penetrates skin.
Beta: electrons and positron. Electrons will chemically react with something very quickly (except in a vacuum, they don’t stick around as free electrons). Positrons will find a nearby electron and be annihilated.
Gamma: very high frequency light. Can be quite dangerous, but doesn’t persist.
None of these transmute other things into radioactive isotopes. The kind of radiation that makes other things radioactive is neutrons, but those are very unusual outside of a nuclear reactor. Fission and some fusion reactions make neutrons, and a couple of radioactive elements make small amounts, and that’s about it. Neutrons also don’t persist in the environment (and, interestingly, they don’t persist very long in space either).
So a spill of hot radioactive sodium is nasty. It’s hot, and it’s highly reactive. But it’s so reactive that it will all react! Sodium can’t meaningfully contaminate groundwater, because it will just turn into salts. It can mess up soil pH, because the reaction product is lye, but that can be remedied by an acid. (Other than its pH, lye is pretty harmless. You use flush it down your drain to clean your drain, and you can even use it to make pretzels!) The radioactive sodium-23 emits radioactive sodium, but much less than 1 trillionth will remain after a day — what’s left is non-radioactive magnesium, which is harmless.
So I wouldn’t want it be around a sodium leak, but visiting it a day or two later while wearing a good pair of boots (for protection against any remaining lye) seems quite safe.
Could an electron emitted by beta decay of a neutron on the sodium hit the nucleus of something else and combine with a proton their to form a neutron, producing an unstable isotope of that something else?
Electron capture by a nucleus is a thing, but as far as I know, this only happens to any significant extent to unstable nuclei that naturally decay that way. It’s probably possible for a high enough energy electron to hit a stable nucleus and convert a proton to a neutron, but I’ve never heard of it (although I’m not an expert). So I don’t think anyone needs to worry about this.
Someone smarter may correct me but my understanding is that radioactivity doesn’t transfer like that. Contamination means for the isotope to find its way into a system (like a human body) and then stay there, radiating harmful particles from the inside.
If you were to be contaminated by ingesting some radioactive sodium, it would still decay and be gone within days.
No, the half-life is a property of a nucleus, which does not take part in chemical reactions.
The normal way of dealing with cases of taking in radioactive abd bio-active elements, like iodine, sodium, (and even strontium which tends to take place if calcium), is taking excessive amounts of the same element, but a normal, stable isotope. Taking in some excessive table salt (cheese and chips anyone?) should be pretty easy.
