Use the "overlaps" operator:

select * 
from  queryables 
where array[users_who_like_ids]  && ARRAY['1','78','80'];

^{Your design is really quesionable: first why are you storing numbers as strings? And secondly why don't you store this as a properly normalized one-to-many relationship?}

Query on array with GROUP BY

To add on to the response by @a_horse_with_no_name, if you want to also GROUP BY against the length of the you need to provide an aggregate function to be used on the grouped values.

SELECT aggregate_function(aggregated_field) 
FROM  queryables 
WHERE array[users_who_like_ids]  && ARRAY['1','78','80']
GROUP BY array_length(users_who_like_ids,1);

For example, I could GROUP BY against the length, and place all the relevant user ids in an array:

SELECT array_agg(queryable_id) 
FROM  queryables 
WHERE array[users_who_like_ids]  && ARRAY['1','78','80']
GROUP BY array_length(users_who_like_ids,1);

I could also GROUP BY against the length, get the average liked_count by array length:

SELECT avg(liked_count) AS avg_liked, 
array_length(users_who_like_ids,1) AS liked_id_count
FROM  queryables 
WHERE array[users_who_like_ids]  && ARRAY['1','78','80']
GROUP BY array_length(users_who_like_ids,1);

Index that array field!

Also, if this table is sufficiently large, lookups will be very slow.

In order to counter this, you will want to index the data in the users_who_like_ids column, but since it is an array field, you cannot simply build a B-Tree index (the default index type), and you should use the GIN index instead:

CREATE INDEX idx_gin_users_who_like_ids
ON queryables
USING gin
(users_who_like_ids);

Then, when applying the array operators &&, @>, or <@, you will find performance to be much better.