PostgreSQL cannot begin/end transactions in PL/pgSQL

Update: after PostgreSQL version 11. you can control transaction inside Store Procedure.

===== Before Version 10:

START TRANSACTION;
select foo() ;
COMMIT;

"Unfortunately Postgres has no stored procedures, so you always need to manage the transaction in the calling code" – a_horse_with_no_name

Transaction in an exception block - how?

A plpgsql function automatically runs inside a transaction. It all succeeds or it all fails. The manual:

Functions and trigger procedures are always executed within a transaction established by an outer query — they cannot start or commit that transaction, since there would be no context for them to execute in. However, a block containing an EXCEPTION clause effectively forms a subtransaction that can be rolled back without affecting the outer transaction. For more about that see Section 42.6.6.

So, if you need to, you can catch an exception that theoretically might occur (but is very unlikely).
Details on trapping errors in the manual.

Your function reviewed and simplified:

CREATE FUNCTION foo(v_weather text
                  , v_timeofday text
                  , v_speed text
                  , v_behavior text)
  RETURNS SETOF custombehavior
  LANGUAGE plpgsql AS
$func$
BEGIN

DELETE FROM custombehavior
WHERE  weather = 'RAIN'
AND    timeofday = 'NIGHT'
AND    speed = '45MPH';

INSERT INTO custombehavior (weather, timeofday, speed, behavior)
SELECT v_weather, v_timeofday, v_speed, v_behavior
WHERE  NOT EXISTS (
   SELECT FROM defaultbehavior
   WHERE  a = 'RAIN'
   AND    b = 'NIGHT'
   AND    c = '45MPH'
   );

RETURN QUERY
SELECT * FROM custombehavior WHERE ... ;

END
$func$;

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If you actually need to begin/end transactions like indicated in the title look to SQL procedures in Postgres 11 or later (CREATE PROCEDURE). See:

• In PostgreSQL, what is the difference between a “Stored Procedure” and other types of functions?