I was lucky that, when this happened, I was ahead on some of my weekly assignments. Unfortunately, I wasn't so lucky on other work, one looming deadline in particular. Plus, I had to miss work at my other job the next day, since I was more or less rendered useless by a fog of Percocet.
Incidents like this highlight the importance of having a contingency plan in place, especially if you work from home. Don't leave things until the last minute, for one thing -- it seems to be Murphy's Law that when you do, something almost always happens to make you late (or even later). Also, it's a good idea to always have some extra money in your account as a buffer, so that if the worst comes to pass, and you miss some work or lose some assignments, you have a cushion to fall back on while you get back on your feet.
If an emergency does occur, make sure you notify your clients as soon as possible if you think there's even a chance it might delay a deadline or make you unavailable for a day or two (or, heaven forbid, more). It might be a good idea to always have a list handy of current clients and their projects, so that if the emergency is so serious that you're not able to contact them yourself, your spouse or another family member or close friend can take care of it for you with minimal hassle.
Do you have a plan in place in case of an emergency? What does yours entail?