From some websites, I found that some of the elements have the value, 0 kJ/mol, in their electron affinities, so, what's that mean?

That means we do not have any information about their affinities or their values are really 0 kJ/mol?  

Don't trust the web.

You know, we are on the web, so our answers can't be trusted too. So when Mitch writes 'Don't trust the web', you should do exactly opposite.

However, if you trust the web, what to do with the Mitch's advice?

Have you heard about Epimenides?

Don't trust the web.

Trust reliable websites. haha. Which websites are reliable - now there's something for you to work out.

If you are not sure what websites, you definitely can't go wrong with books, aka literature data.

Such books include the infamous CRC Handbook for Chemistry and Physics.

I have never seen cases of 0 kJ/mol for electron affnity values.

When a species accept electrons, its enegy content either increases or decreases at fixed pressure and temperature. It is not possible for 0 kJ change in terms of its energy content. When an electron-rich species accept electron, there is increased electronic repulsion within the specieis, thus destabilising the species. When an electron-poor species accept electron, the attraction between the nucleus and the donated electrons lower the energy content of the species, thus lowering its energy content.