i suppose it is possible but your child may e out of diapers by the time you are done making them. you could get an inexpensive sewing machine cheaper than what using disposables will end up costing you and besides making diapers you can make all sorts of other clothes so a sewing machine would end up paying for itself. it is so easy and fast to make toddler pants out of old shirts and things since i started sewing we haven't had to spend any money on new clothes for my son i just take old clothes from myself and my husband and use the fabric to make new toddler sized clothes.
mama to two amazing children son 10/27/07 and daughter 07/07/11
If you don't mind the frayed edge look, you don't have to hand sew anything at all... Just cut out the shape of a diaper for the outer layer that will wrap around baby. The inside soaker can be a large rectangle of fabric that you fold up. Pin 'er up and you are good to go!
Also, you could just sew prefolds (so basically a rectangle of several layers) and pin them. Obviously this could get a bit "damp" if you didn't make a cover (can you knit or crochet? or alter old wool sweaters by hand which is also pretty quick/easy to do)... if you do have a well fitting cover you could even skip the pinning.