Is "Fish fish fish eat eat eat" a grammatically correct sentence?

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It is. The sentence means something like "Fish that are eaten by fish, which are in turn eaten by fish, eat." To see why this is grammatical, we can look at the set of sentences below.

Fish eat.

This is obviously grammatical. Now, we modify the matrix subject with a relative clause. I have put the word being modified in bold.

Fish eat. > Fish [fish eat] eat.

This may be a little confusing, but compare the structure of the sentence above with the one below:

People eat. > People [I know] eat.

We can see that it is the same structure. That is, the subject (fish and people, respectively) is the object of the relative clause (i.e. of eat--the one inside the relative clause--and know, respectively).

Then, we take a step further and do the same again with the subject of the relative clause. Now we have another relative clause inside a relative clause.

Fish [fish eat] eat. > Fish [fish [fish eat] eat] eat.

This can actually continue ad infinitum, as long as the numbers of fish and eat are the same.

• fish [fish [fish [fish eat] eat] eat] eat
• fish [fish [fish [fish [fish eat] eat] eat] eat] eat
• fish [fish [fish [fish [fish [fish eat] eat] eat] eat] eat] eat
• fish [fish [fish [fish [fish [fish [fish eat] eat] eat] eat] eat] eat] eat
• ...

You can see that the brackets that show what modifies what come towards the center of the sentence. This is why linguists call it center embedding. Center embedding is known to be difficult to parse, despite its grammaticality. In this case, it's even more difficult because it consists of only two words.