We have a large steel bowl that is easy to get to which is the sick bowl, just as my parents did, might be the same one! If you need to be on the toilet as well, then the bath is okay to be sick into although it can be tricky to clear out afterwards. Or a washing up bowl. Try to keep the sink free for washing your hands. In fact, anything that can be cleaned with very hot soapy water is fine to be used in an emergency to avoid making a worse mess. Bleach is not strictly necessary although you can use that on some surfaces if you want to.
I had to sacrifice an old towel to cleaning up a huge pile of sick on the floor once, double bagged it and took it out in the morning. But it was fine. We try not to get it on the bedding or the carpet if possible as any of the hard floors are easier to clean, but carpet shampoo and then later, Febreeze, should sort it. No need to boil wash the linens, most of which wouldn't tolerate 90 degree washing anyway. 40 degrees with a prewash or extended wash and a decent amount of laundry detergent is fine. 60 if you think the bedding will tolerate it and 60 for any towels that were involved. Use biological powder if you can tolerate it, and that laundry sanitiser if you want to, but neither is necessary.
I have long hair so I always have bands to hand to tie it back, there are little dishes of them all over the place in my house not specifically for sick incidents! Agree with rehydration salts, sipping these, and then plenty of fluids and very plain food for a day when you are ready, and then build up.
A friend has carpet tiles in her children's bedrooms, you can just lift these up and rinse them down. You could even get some of that sawdust they use in schools if you want, to have to hand.
I am not that prone to being sick but a medical condition was making me more prone to food not agreeing with me last year, hence the above. Plus have had sicky children over the years. Car sickness is a different topic again!