What article should be used in the following sentence?
He was English by [a/the/] blood.
I feel there should be a zero article here, but I was taught that the zero article is impossible in English.
He was English by blood is correct.
It isn't true that the zero article is impossible in English. It all depends on the sentence.
No article is required here because it's a fixed expression indicating family relationship by birth rather than marriage. Example: They are related by blood. My explanation is based on CALD3.
Who taught you that there is no zero article in English?
Briefly, articles work like this:
Singular countable noun ~ ~ definite - the ~ ~ indefinite - a/an
Plural countable noun: ~ ~ ~definite - the ~ ~ indefinite - zero
Uncountable: ~ ~ ~ ~ ~ ~ ~ ~definite - the ~ ~ indefinite - zero
In detail, they work like this.