..and even without insulin many of your cells can access some glucose. The muscle/skeletal cells in particular have 2 forms of receptors, one activated by insulin, one by muscle contraction, so when you move it's really very easy for insulin to get into the cell. Thats why many of us must eat/ reduce insulin prior to exercise.
On th other hand, without any insulin, your liver would attempt to continue to supply you with glucose. Insulin acts a bit like a brake on this function. It also acts to control the breakdown of fats mentioned by the poster above. If there isn't any insulin, then that fat will continue to breakdown, producing ketones regardless of food intake, or energy need. Without any insulin , It is possible to be literally starving (little or no food) and even have relatively low glucose levels and still develop DKA. There are young people in developing countries diagnosed in this state.
Many T1s, particularly if diagnosed beyond childhood, also still produce a tiny amount of their own insulin. This can help delay problems a bit.