There are infinitely many solutions but the smallest is that a, b, c and d are (in some order) -11,-2,7 and 10.

Their sum is 4, and the sum of their cubes (-1331, -8, 343, 1000) is also 4.

I arrived at this answer partly by narrowing the scope of where I was looking, and thereafter trial and inspection. I remembered that ALL cube numbers are either 0,1 or 8 modulo 9 (in other words they all cube numbers are either a multiple of 9, or one away either side). You can verify this astonishing fact by just trying the first nine cube numbers and observing that this is in fact true for those, and because we are working in module 9, it is therefore true for all cube numbers.

For our four cube numbers to add to 4, they must all be of the +1 variety. This happens whenever the number to be cubed is 1 more than a multiple of 3, ie: 1, 4, 7, 10, etc and -2, -5, -8, -11, etc. So we only need to look at the cubes of these numbers. Obviously the trivial 1 appears in this list, as do each of the numbers in my answer: -11, -2, 7 and 10.