How to exclude NULLs from ARRAY so query won't fail

Glad you asked! BigQuery supports IGNORE NULLS and RESPECT NULLS modifiers in some of the aggregate functions, including ARRAY_AGG, so your query becomes

#standardSQL
SELECT ARRAY_AGG(x IGNORE NULLS) FROM UNNEST([1,NULL,2,3]) x

and it passes producing [1,2,3]. More details are in the documentation.

Another interesting use-case would be if you do not want to lose that NULL elements but rather want to substitute it with some default value. For example -999

Below will do this

#standardSQL
SELECT ARRAY_AGG(IFNULL(x,-999)) FROM UNNEST([1,NULL,2,3]) x   

And in case if you want distinct elements only -

#standardSQL
SELECT ARRAY_AGG(DISTINCT IFNULL(x,-999)) FROM UNNEST([1,NULL,2,3,1,NULL]) x

Complementing @Mikhail Berlyant's answer, sometimes you want to keep the NULL values somehow but can't use a placeholder. For example, if you want a boolean array with NULLs, where you can't substitute NULL for true/false.

A nice workaround is wrapping the values inside a STRUCT, then constructing the array:

#standardSQL
SELECT ARRAY_AGG(STRUCT(x)) AS struct_array FROM UNNEST([1,NULL,2,3]) x