Is there a way to set an "expiry" time, after which a data entry is automatically deleted in PostgreSQL?

There is no built in expiration feature but if your goal is to automatically expire fields and have the logic contained within your database (and thus no outside dependency like a cron job) then you can always write a trigger. Below is an example of a trigger that deletes rows from a table that have a timestamp of older than 1 minute. It is executed whenever a new row is inserted into that same table. You can obviously set the trigger to execute on other conditions and for various expiration dates as need be. I used the following website as a basis for this: http://www.the-art-of-web.com/sql/trigger-delete-old/

CREATE TABLE expire_table (
    timestamp timestamp NOT NULL DEFAULT NOW(),
    name TEXT NOT NULL
);

INSERT INTO expire_table (name) VALUES ('a');
INSERT INTO expire_table (name) VALUES ('b');
INSERT INTO expire_table (name) VALUES ('c');

select * from expire_table;
         timestamp          | name 
----------------------------+------
 2014-09-26 15:33:43.243356 | a
 2014-09-26 15:33:45.222202 | b
 2014-09-26 15:33:47.347131 | c
(3 rows)

CREATE FUNCTION expire_table_delete_old_rows() RETURNS trigger
    LANGUAGE plpgsql
    AS $$
BEGIN
  DELETE FROM expire_table WHERE timestamp < NOW() - INTERVAL '1 minute';
  RETURN NEW;
END;
$$;

CREATE TRIGGER expire_table_delete_old_rows_trigger
    AFTER INSERT ON expire_table
    EXECUTE PROCEDURE expire_table_delete_old_rows();

INSERT INTO expire_table (name) VALUES ('d');

select * from expire_table;
         timestamp          | name 
----------------------------+------
 2014-09-26 15:36:56.132596 | d
(1 row)

No. There is no such feature.

I can't see what it does more than either (1) just an "expired" timestamp does or (2) timestamp + cron-job/pgAgent.

It doesn't sound like a general feature that would be added to the core. You could quite simply code an extension to handle this sort of thing, with either a tick called from a cron-job or perhaps a background-worker process.

I don't see anything on pgxn, so presumably there's not been much demand for it yet.