[susan53]

The second version of the question is unnatural - we always say How old and not How many years old. As always, I've checked with a concordancer and there are no examples at all of How many years old in the corpora.

Either answer is acceptable however, and should be marked right if it's a comprehension question. In spoken English, it would be more likely for Three years old to be prefaced by he was however. You could say :

A : How old was John... ?
B : Three or He was three.

or
A: How old was John...
B: He was three years old or (possibly but less likely) Three years old

I checked a concordancer of spoken English and in 15/16 examples of British English the expression XX years old was preceded by the verb BE. In American English BE was used in 6/9 examples. So if conversational skills are being tested (and in that case why is it a written test) either is possible but He was 3 years old is more natural.