Postgres SQL Exclusive OR (XOR) CHECK CONSTRAINT, is it possible?

You can't compare NULL values with =, you need IS NULL

(a IS NOT NULL AND b is NULL) OR (b IS NOT NULL AND a is NULL)

For a check constraint you need to enclose the whole expression in parentheses:

create table xor_test 
(
  id integer primary key, 
  a integer, 
  b integer, 
  check ((a IS NOT NULL AND b is NULL) OR (b IS NOT NULL AND a is NULL))
);

-- works
INSERT INTO xor_test(id, a, b) VALUES (1, null, 1);

-- works
INSERT INTO xor_test(id, a, b) VALUES (2, 1, null);

-- fails
INSERT INTO xor_test(id, a, b) VALUES (3, 1, 1); 

Alternatively the check constraint can be simplified to

check ( num_nonnulls(a,b) = 1 )

That's also easier to adjust to more columns

Right, the a = NULL and b = NULL bit was the issue as @a_horse_with_no_name indicated. You might also consider this derivative, which doesn't require the OR operator:

create table test 
(
  id integer primary key, 
  a integer, 
  b integer, 
  check ((a IS NULL) != (b IS NULL))
);

Of course that works exclusively with only two column XOR comparison. With three or more column XOR comparison in a similar test table you could resort to a similar approach more like this:

create table test 
(
  id integer primary key, 
  a integer, 
  b integer, 
  c integer, 
  check ((a IS NOT NULL)::INTEGER + 
         (b IS NOT NULL)::INTEGER + 
         (c IS NOT NULL)::INTEGER = 1)
);