Edit: Interesting to see this get voted up and down. I suppose because it appears to have a holier-than-thou tone. I don't really feel that way because I struggled to learn this stuff the hard way. I'm just hopeful someone will feel encouraged to try it.
Not the only one. Admittedly I don't have kids, but I'm familiar with family life and I don't think this is really that much of a problem.
I think a lot of people suffer from the problem of not being educated about how to do basic things around the house. They make huge problems for themselves. For dishes, the main key is just not to ever have dirty dishes. Definitely, do not store dirty dishes in the sink. Not only does it stop you from cooking, but it means you are almost certain to break stuff when you do eventually wash it (because there is no room).
The idea of storing dirty dishes in a dishwasher is also just bad. You either need 10 of everything, or you are stuck digging around and hand washing some horribly encrusted pot that you need. Instead, as you cook, clean. If you are in a typical North American kitchen, box off 90% of it and constrain yourself to an area that includes the sink, an area the size of your cutting board and your stove/oven. Hopefully the sink, board and stove are right next to each other. Keep a small dish of hot water and detergent right next to the sink with a sponge or cloth in it. Replenish occasionally (I usually use the kettle since it is faster and more efficient than trying to get small amounts of hot water out of the tap -- I lived 5 years in a house without hot running water and never missed it!).
Whenever you use anything, clean it right away. Also put it away. Specifically: pots, knives and cutting boards should never be dirty. Get a knife board (or similar). Clean your knife and put it away every time you use it. The 2 seconds you save by putting your knife on the board (or other hazardous place) is not worth it. Pots should be cooking something, storing something, or spotless. Similarly, your board should be in use or clean.
Try to make due (or 95% of what you do) with a frying pan, 1 small pot and one large pot. Then reuse these as you are cooking. Also, buy 1 good knife and do everything with it. Think of these things as being in your cache. Make it a cache miss to swap out to something else. Obviously you will occasionally need specialised tools, but try to keep these really infrequent. Finally, wash the kitchen before you serve the meal. It sounds crazy, but you should be able to do it in less than 5 minutes, so the food can wait. Not only will your kitchen always be clean, but you will dramatically increase the speed at which you can cook. It is really astounding how much better it is.
At that point, the only thing you ever need to wash are the dishes you are eating off of. If you are a couple, then it takes virtually no time. If you are a family, then this is good "family time" (teaching children that working is part of life is not the most terrible thing you can do).
For laundry, my wife does the laundry every day that's sunny. We store dirty laundry in the machine. (Hint: Always leave the machine door open if you can. A dry machine is a not-moldy-machine). As soon as we get up my wife turns on the machine. After breakfast she hangs the laundry outside. Total time: 5 minutes. When she gets home at the end of the day she folds the laundry. I work from home so if it starts to rain, or is too windy, I take the laundry in and fold it. Total time: 5 minutes.
When we lived in London (i.e. a-not-sunny-place), we hung everything indoors -- just make sure that the room is well ventilated, or you can cause problems with moisture. It sometimes took 2 days for some things to dry thoroughly, but you can quickly put a few things in the dryer for a couple of minutes if you don't want to wait -- normally takes less than 5 minutes to finish it off.
I used to wash all the laundry by hand (really)! Before breakfast, soak. Before taking a shower, drain and re-soak with detergent. After having a shower (not having clothes on means you don't have to worry about getting wet), wash and wring out the detergent. Rinse and soak in rinsing water. After getting dressed, brushing teeth, etc, wring out the water, then lay the clothes out onto towels. Roll up the towels with the clothes inside. Twist the towels to transfer the water into the towels. Total time: about 10 minutes. Downside: you need a couple of extra towels that you need to hang.
Now, the above is hard work if you have a lot of clothes, but it's practically trivial if you are doing it every day (I still do this when travelling). To be honest, a modern washing machine is stupidly efficient and does quite a good job, so might as well use it, but it's useful to realise how little hassle it is to do it by hand -- as long as you are doing it every day!