It is trival to see that n<4 is not possible. 

Suppose that we have some nice set S with an even
number n of elements.  For each n-1 subset of S, we
see that the sum must be even.  Let the sum of subsets
be s, an even number.  Each number is counted n-1
times.  So the sum of elements in set S is s/(n-1),
which certainly is an even number since n-1 is odd.
Immediately the difference between sum of elements
s/(n-1) and every n-1 subset sum is even (difference
between two even numbers).  But this difference is
just every element in the set.  ie every element is
even is a necessary condition.

But if every element is even, then we can keep
dividing by two and see that at least one of the gets
to be odd, and our original hypothesis is still true.
but this contradicts the fact that every elements must
be even.  Hence no nice set S of even cardinality
exists.

Now we will prove n=5 does not work.  let its elements
be a,b,c,d,e.  WLOG assume a<b<c<d<e
consider the subset {a,b,c,d} (or any other 4 subset)
because a<b<c<d, then a+b+c=d is possible.  obviously
a+b<c+d, a+c<b+d, hence the other case is a+d=b+c
case 1:
a+b+c=d
but then in {a,b,c,e}
e>a+b+c hence this is not feasible
then case 2:
a+d=b+c
in {a,b,c,e}
then we must have
a+b+c=e
hence {a,b,c,d,e}={a,b,c,b+c-a,b+c+a}
but in {a,b,b+c-a,b+c+a}
we must have 2b+c-a=b+c+2a
b=3a
{a,b,c,d,e}={a,3a,c,c+2a,c+4a}
removing a,
c+7a=2c+5a
c=2a
a contradiction
so n=5 is impossible.

Finally we show that n=7 works by an example: 1 3 5 7
9 11 13
 

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