Based upon your description, it sounds as though you have most likely just bruised the tissue covering the spinous processes of the vertebral column. This is easy to do if you are performing your sit-ups on a hard floor, as there is no muscle and little fat to cushion your mid-line as you set yourself down between each repetition. If it is just bruising, there is no danger in continuing with your training.
And yes, the best solution is an exercise/yoga mat or a sit-up bench. Alternatively, just a towel folded over twice tucked under the ischial tuberosities (sit-bones) can offer enough padding to elevate the hip and relieve the pressure on the lumbar spine.
To be certain that it is just bruising, palpate the skin along the vertebral column. If it is, the skin will feel tender directly on top of the spinous processes. If there is pain or discomfort between or lateral to the processes, or deeper within the vertebral column, you should seek advice from a physiotherapist or sports doctor.
I hope that is helpful.