Would space feel cold?

The answer is both!

The background temperature is colder in space than on Earth because we have an atmosphere, which keeps us warm. So at the distance the Earth is from the Sun, it's about -50 degrees Celsius in space.

On Earth, we lose heat through radiation, evaporation of sweat, and by conduction of heat to the air. It is harder to lose heat in space as you can only lose heat through radiation. This is where electrons in the molecules in our body go from a high energy state to a lower energy state, losing energy when they do so by radiating infra-red (heat) radiation.

Conversely, when you face the Sun, you will absorb the Sun's radiation and be warm. But, at the same time, your back will be in shade and will lose heat rapidly by radiation as there's no atmosphere to keep you warm.

In other words, large temperature gradients are found in space, so things get very hot when they are in the Sun's rays, but very cold when they are in shade, so the answer to the question is space feels both cold and warm!