Hi Wayne Yes, this can be done....but at considerable expense.
As Jim points out, your house will have to be raised off the existing foundation. If you are in a warm climate, the footings will no doubt not extend deep enough to go under the basement floor. If in a colder climate, basements or cellars are common, because they have to excavate down so deep for the footings that its just as easy to install the basement while they are there. Otherwise, you will have to break up the existing poured or cinderblock walls and footings, extract all pieces and then dig out the basement cavity. If the main sewer pipe exits above the new basement floor level, you will have to install a lift pump for any plumbing you install in the basement. Then forms will have to be made, concrete pumped into the forms and the basement slab, the concrete cure for at least 3 weeks and then the house reset and plumbed on the new foundation. Whew....a lot of work and if you pay for someone to do it start to finish....well...it better be a very valuable house on very valuable property to make it worth it :-) Best