Learn Dutch: Alle mensen hebben goed voedsel nodig.

Because it is talking about people in general, rather than a specific group. You use alle plus a plural noun for general statements. If you were talking about a specific, defined group, like 'all the people in this room', you would use al de mensen.

Why is it goed voedsel without an -e on the end of the adjective?

Voedsel is a het-word (het voedsel). When you put an adjective before a het-word that has no article (or has een before it), the adjective does not get an -e ending. So it remains goed instead of goede.

Why is nodig placed at the very end of the sentence?

It belongs to the verb combination nodig hebben, which means 'to need'. In a normal main sentence, the conjugated verb (hebben) takes the second position, and the other part (nodig) gets pushed all the way to the end of the sentence.

Could I use eten instead of voedsel here?

Yes, eten also means 'food' and is perfectly fine to use. Voedsel sounds slightly more formal, scientific, or general—similar to talking about 'nutrition' or the global food supply—which fits well with a broad statement about all of humanity.