That's why having a fully stocked first aid cupboard with all types of bandages and other first aid supplies is a good idea. Having everything in one special place is a lifesaver because when someone is sick or injured you don't want to have to search around to find something.
My personal home kit is organized with compartments to separate things into. All I have to say to the kids is "Go to the first aid kit" and off the they run to the cabinet. Even when my kids were as young as 4 they knew where it was and how to get it.
I have my kit separated into several sections. The biggest one is for bandages, first aid tape, gauze pads, wrapping tape, and ace bandages for sprains. The three small ones are separated into eye first aid (flushing drops, antibiotic drops, allergy drops), ointments (triple antibiotic, hydrocortisone, fungal, burn crème), and medical implements (thermometers, tweezers, nail clippers, bulb syringe, lice comb). Next to the kit are bottles of rubbing alcohol, hydrogen peroxide, iodine, and witch hazel. I also have a big bag of Cotton balls for cleansing wounds. Above the kit (out of small children's reach) are medications. I always keep i.b. profen, Tylenol, Benadryl, cough meds, vapor rubs, allergy meds, cough drops, antidiarrheal, antacids, and essential oils on hand and in infants, children's and adult strength. I also keep a few 6 packs of Gatorade on hand at all times in case of illness. I buy my meds on sale and more than one at a time because generally speaking when one child gets sick they all get sick and then I get sick. So we go through allot of meds at once. There is nothing worse than taking your sick child to the store to pick up a needed medication. This saves me from participating in that kind of insanity. Every six months I go through and check expiration dates.
Also a consideration is if you couldn't get to the store due to bad weather or natural disaster. The last thing I want is to have someone get sick and have no way to help them.