Today, a small load can be washed in the kitchen or bathroom sink in less than a half an hour. Large items, like blankets or comforters, can be washed in the bathtub.
Cold water works best to remove stains. Have about twice as much water as clothes, add laundry detergent, agitate (push the water through the material) for about 5 minutes, then let soak for 5 minutes. Agitate again for 5 minutes. Drain all water from the sink and individually rinse each item. Wring out and hang to dry.
As the clothes will drip while drying, choose the drying location carefully. An outdoor clothesline is ideal, if you have one and if weather permits. A folding wooden drying rack in the middle of the bathtub is optimum, as the drips will go directly down the drain. The shower curtain rod is a great option as well for the same reason. In a pinch, drape the clothes over the side of a laundry basket placed in the sink or tub.
Once dry, vigorously shake out each piece to remove any lint. If you do a hand wash in the evening, it will usually be dry by morning. Relocate damp items from the shower curtain rod to the towel rack or even the back of a chair if needed.
Total savings? The cost of a wash/dry at the laundromat, plus transportation costs to get there and home. Depending on where you live, that can add up to quite a bit over time.