The topic of sanitation covers everything from after-meal cleanup to human waste. I can tell you what works for us but I can't tell you what the best solution is for you. Too much depends on where and how you live. The main thing is to have a plan that will work for you.
When the faucet runs dry, even washing dishes can be tedious and time-consuming. This is where stockpots come in handy as you can keep large quantities of water warming on the woodstove or other heat source. Make sure you have a couple of sturdy dishpans, a good draining rack, and plenty of dish soap. You'll also need scouring pads, dish cloths, and dish towels. Save your used dishwashing water to throw on your garden plants as it can keep many kinds of garden pests from taking up residence on your vegetables. If you've done a good job washing your dishes, your rinse water should be clean, albeit soapy, and you can re-use it to wash your socks.
You can wash most if not all of your clothes in the sink, being careful not to over-suds them which can make rinsing a water-consuming task. If you have a woodstove, you can hang a drying rack near the ceiling and your clothes will dry quickly. A set of clothes bars (standing drying rack) will be most helpful for both regular laundry and drying off clothes you've worn out in the snow or stream. In dry weather you can hang your clothes on the line to dry - don't forget a supply of clothespins. You can still get washboards or, for those getting-off-the-grid folks, a manual wringer washer might do the trick. Rolling your shirts, sweaters and jeans up in a dry towel will get most of the excess moisture out prior to hanging. Since I liked to get the wrinkles out before going to work, I bought several bottles of wrinkle-remover spray. You can accomplish the same thing with diluted liquid fabric softener in a spray bottle. Just spray wrinkled areas and hand smooth them, then let hang to dry. The same thing goes for wash water that applied to dish washing water - save it for your garden! Rinse water can be re-used as wash water to conserve supplies.
If you think you can't live without a shower you're wrong - you don't even need a bathtub. What you do need is a warm room, some hot water, soap, a washcloth and a towel. If you like to rinse your hair with running water, buy a couple of square water jugs with spigots and stick your head under them. In the summertime you can use a sun shower if you're in a area where you can avoid passing eyes! If you have a standing shower you can also attach a hand-shower to the bottom of a bucket, being careful to seal around the seam very carefully with heat-resistant caulk. Keep water heating all the time on your woodstove if you have one, or keep a big stockpot filled and ready to heat on your alternate cooking stove. While this may not be height of luxury in the bathing world, it's perfectly effective and your friends won't know the difference. Wash your hair at night and let it dry before bedtime. You can use a misting bottle to take care of the morning frizzies and a butane curling iron if your locks need some beautifying!
When planning your long-term supplies, think in terms of compost piles and burning barrels. You can keep your garbage disposal problems at a minimum if you avoid having lots of small plastic bottles. Buy supplies in large plastic containers or in boxes; e.g. soaps and other daily-use items should be purchased in the largest possible container and then transferred to smaller containers for ease of use.
If you have a woodstove you can use waste cardboard and other paper for fire-starting. Just make sure you keep the creosote cleaned out of your chimney. Periodically let the fire die out and the pipes get cold. Bang the pipes with a small stick or metal handle to knock down the large chunks of creosote. When you re-start your fire, let it burn hot for about 20 minutes to burn the rest of the creosote from the pipes - once a week should do it if you're using your woodstove every day. This procedure may not be according to Hoyle, but I know from experience that it works since I never burned the cabin down even with daily use of the woodstove.
While burning papers in your woodstove or in a burning barrel may not be politically or environmentally correct, in an emergency situation some laws have to be carefully weighed against survival considerations. Make sure you have a fine grate over your barrel to keep burning ashes from starting grass fires.
One of the most useful garbage disposal tools you have is a compost pile. All vegetable matter can be composted into excellent garden fertilizer. The heat generated at the bottom of the compost heap as rotting occurs accelerates the composting process throughout the pile. You can make a compost pile using light fencing wire and some metal fence posts, forming a circle of wire which can be easily opened when it's time to turn the pile. Put some grass clippings, leaves, small branches and brush at the bottom of the pile, then add your food garbage to the pile, adding more grass and other greenery as you accumulate it. A compost pile doesn't stink, but it may attract unwanted guests in the form of wild animals. Unless you live in bear or moose country, you don't need a solid enclosure for your compost pile. How composting works can be visually illustrated by filling your compost pile to the top of the fence and then checking back in a few days - the pile will be 1/3 to 1/2 the size it was when you left it. Open the fencing and turn the pile over every few months to get the most effective composting. Compost will render your normal garden soil rich and fertile.
If you have garbage that won't fit into either of the above categories for disposal, you'll probably need to find a way to create your own landfill. There are places in the country where you're allowed to have a small landfill on your own land. Be careful about what you put in your landfill. I would suggest that it not extend beyond plastics and clean metal cans. When the crisis is past you can gather up your junk and take it to the regular collection station. In the interest of reducing the size of your landfill, avoid small plastic containers and materials that aren't prone to easy disposal.
All of the above is based on my own experience at - you guessed it - the cabin. I had very little "hard" garbage and could generally fit it all into one 13-gal. trash bag every month. Learning how to keep things clean and take care of your trash is important, since you won't want to be figuring this out after you're already buried in a mound of junk! Now for some more benefit of my experience...
Alternatives to the Flush Toilet
You can get a chemical toilet, a composting toilet, or you can build yourself a composting outhouse. Deciding factors center around where you live and what you can live with.
While the notion of an outhouse may turn you a pale green, they aren't quite as bad as you might think if properly cared for. You'll need about 50 lbs. of lime per year plus some ashes from your wood fire. You can throw in some grass clippings from time to time to aid in the composting process. In the winter months, you might want to hang a toilet seat cover near the woodstove so you can caryy a warm seat to your outhouse! The one I had at the hunting cabin was a 2-holer complete with regular padded toilet seats. I limed it after every use and added grass clippings every time I mowed the yard (with, of course, my manual push mower). The cabin was set up with a main room and bunkroom, attached garage and woodshed, with the outhouse accessible through the back of the garage. Did that make it an in-house? Whatever the case it was infinitely better than having to trudge through the snow in the middle of the night.
Chemical toilets or commercial composting toilets are much neater and better smelling than outhouses, but disposal of waste products is an extra consideration. You can find these toilets at RV dealers or from Lehman's or other sources. For short-term use these toilets are fine; make sure you have enough chemicals on hand to get you through an emergency situation.
Links to chemical toilets
How to Build a Simple Outhouse