Denormalized method (do not do this)

First, we create the first two tables

CREATE TABLE people    ( pid serial PRIMARY KEY, name text, contact text );
CREATE TABLE questions ( qid serial PRIMARY KEY, question text );

One method of doing this is just expanding the check statement to satisfy all permissible variants.

CREATE TABLE answers (
  pid       int  REFERENCES people,
  qid       int  REFERENCES questions,
  option_a  bool NOT NULL DEFAULT false,
  option_b  bool NOT NULL DEFAULT false,
  option_c  bool NOT NULL DEFAULT false,
  option_d  bool NOT NULL DEFAULT false,
  PRIMARY KEY ( pid, qid ),
  CHECK (
       (    option_a AND NOT option_b AND NOT option_c AND NOT option_d)
    OR (NOT option_a AND     option_b AND NOT option_c AND NOT option_d)
    OR (NOT option_a AND NOT option_b AND     option_c AND NOT option_d)
    OR (NOT option_a AND NOT option_b AND NOT option_c AND     option_d)
  )
);

That can be simplified a bit.

CREATE TABLE answers (
  pid       int  REFERENCES people,
  qid       int  REFERENCES questions,
  option_a  bool NOT NULL DEFAULT false,
  option_b  bool NOT NULL DEFAULT false,
  option_c  bool NOT NULL DEFAULT false,
  option_d  bool NOT NULL DEFAULT false,
  PRIMARY KEY ( pid, qid ),
  CHECK (
       (    option_a AND NOT   option_b AND NOT option_c   AND NOT option_d)
    OR (NOT option_a AND (     option_b AND NOT option_c   AND NOT option_d)
                     OR  ( NOT option_b AND (     option_c AND NOT option_d)
                                        OR  ( NOT option_c AND     option_d)))
  )
);

Alas, none of this is normalized. To normalize this design just do this..

CREATE TABLE answers (
  aid           serial PRIMARY KEY,
  description   text
);
CREATE TABLE question_answers (
  pid    int  NOT NULL REFERENCES people,
  qid    int  NOT NULL REFERENCES questions,
  aid    int  NOT NULL REFERENCES answers,
  value  bool NOT NULL DEFAULT false,
  PRIMARY KEY (qid, aid, pid)
);

Normalized Method

Now, if you wish to stop more than one value from being true in the above you can use a PARTIAL UNIQUE INDEX

CREATE UNIQUE INDEX
  ON question_answers(pid,qid,aid,value)
  WHERE value = true; -- (or just WHERE value)

First one seems fine, I would save if it is correct on the option though.