How to not "duplicate" a table if it has a unique constraint over many columns?
With PostgreSQL 9.5 you can enjoy a BRIN index (which will make the index very small, yet functional), and handle the exclusions via a trigger, like this:

CREATE INDEX ON medi_cal_base_eligibility USING BRIN (client_index_number);

CREATE OR REPLACE FUNCTION tf_medi_cal_base_eligibility_insert() RETURNS trigger AS
$BODY$
BEGIN
  IF (TG_OP = 'INSERT' OR (NEW.client_index_number, NEW.eligibility_date, NEW.medi_cal_date, NEW.cardinal) IS DISTINCT FROM (OLD.client_index_number, OLD.eligibility_date, OLD.medi_cal_date, OLD.cardinal)) AND
     EXISTS (SELECT 1 FROM medi_cal_base_eligibility
             WHERE (client_index_number, eligibility_date, medi_cal_date, cardinal) = (NEW.client_index_number, NEW.eligibility_date, NEW.medi_cal_date, NEW.cardinal)) THEN
    RAISE 'Duplicate key: %, %, %, %', NEW.client_index_number, NEW.eligibility_date, NEW.medi_cal_date, NEW.cardinal;
    RETURN NULL;
  END IF;
  RETURN NEW;
END;
$BODY$ LANGUAGE plpgsql VOLATILE SECURITY DEFINER;

CREATE TRIGGER t_medi_cal_base_eligibility_insert BEFORE INSERT OR UPDATE ON medi_cal_base_eligibility
    FOR EACH ROW EXECUTE PROCEDURE tf_medi_cal_base_eligibility_insert();


As stated by dezso, a BRIN index would only be useful in a situation in which client_index_number has a correlation to the position of the record in the file.

If the solution above using BRIN won't cut the mustard, a good choice is using a hash of the data to search for. The size of the hash will determine how many records it will have to scan to look for uniqueness; also, the bigger the hash, the bigger the index. A hash of 32 bits would most likely render unique results (or at most a handful) and would be just as large as your primary key. In the following example I'm getting the 32 bits hash by using the last 8 hexadecimal digits of the md5 function using your four unique fields concatenated together.

CREATE OR REPLACE FUNCTION f_medi_cal_base_eligibility_to_int (p_client_index_number text, p_medi_cal_date date, p_eligibility_date date, p_cardinal smallint) RETURNS int AS $BODY$
  SELECT ('x'||right(md5($1 || to_char($2, 'YYYYMMDD') || to_char($3, 'YYYYMMDD') || $4::text), 8))::bit(32)::int
$BODY$ LANGUAGE SQL IMMUTABLE SECURITY DEFINER;

CREATE INDEX ON medi_cal_base_eligibility (f_medi_cal_base_eligibility_to_int(client_index_number, medi_cal_date, eligibility_date, cardinal));

CREATE OR REPLACE FUNCTION tf_medi_cal_base_eligibility_insert() RETURNS trigger AS
$BODY$
BEGIN
  IF (TG_OP = 'INSERT' OR (NEW.client_index_number, NEW.eligibility_date, NEW.medi_cal_date, NEW.cardinal) IS DISTINCT FROM (OLD.client_index_number, OLD.eligibility_date, OLD.medi_cal_date, OLD.cardinal)) AND
     EXISTS (SELECT 1 FROM medi_cal_base_eligibility
             WHERE (f_medi_cal_base_eligibility_to_int(client_index_number, medi_cal_date, eligibility_date, cardinal), client_index_number, medi_cal_date, eligibility_date, cardinal) =
                   (f_medi_cal_base_eligibility_to_int(NEW.client_index_number, NEW.medi_cal_date, NEW.eligibility_date, NEW.cardinal), NEW.client_index_number, NEW.medi_cal_date, NEW.eligibility_date, NEW.cardinal)) THEN
    RAISE 'Duplicate key: %, %, %, %', NEW.client_index_number, NEW.medi_cal_date, NEW.eligibility_date, NEW.cardinal;
    RETURN NULL;
  END IF;
  RETURN NEW;
END;
$BODY$ LANGUAGE plpgsql VOLATILE SECURITY DEFINER;

CREATE TRIGGER t_medi_cal_base_eligibility_insert BEFORE INSERT OR UPDATE ON medi_cal_base_eligibility
    FOR EACH ROW EXECUTE PROCEDURE tf_medi_cal_base_eligibility_insert();