Rights at work
It is a good idea to look into your rights at work when it comes to caring for a sick child or other relative. Some countries, like the United Kingdom, give employees the right to compassionate leave by law to be able to do this. Other countries do not protect employees in the same way, meaning you may have to negotiate with your employer, take holiday time, or look into other options. Either way, it can be really helpful to know where you stand before you ever have a sick child, so you don't have to make last minute arrangements.
What if you can't take time off?
Parents who have no way to take time off work find themselves in a tricky situation. Owning your own business makes for an especially complicated recipe. If you and your partner (if you have one) can't stay home from work, perhaps a relative or friend can do this for you. In some cases, you may be able to at least reduce your working time, or finish a task at home. If you know in advance that taking time away from work is hard, make sure you have an emergency plan in place once again, before your child gets sick.
Working from home
I'm a self-employed work-at-home mom, and though I work very part time, I do have stuff to do on a regular basis. Working at home and caring for a sick child can seem like a nightmare. Actually, my four-year old is sick as I write this and there are ways to make it work. She has a fever and is currently sleeping, so I have some time to finish my work. The rest of the day is free to care for her.