I have one child, so my information isn't so much about what to do with four children while you are sick. I wouldn't have a clue. LOL However, I try to think of the big picture for these types of things. What lesson or overall experience do I want my child to learn about this "situation"? For me, it has changed over the years as my child has gotten older (age 7 now). I also balance how effective I am as a mother at that given moment. Are my needs being met? Is my child physically safe?
Personally, I see nothing wrong with popping in a DVD here and there. I rarely choose to do it myself, but it is because my child is not entertained by movies or TV and I don't rest well when it is on.
My advice is to trust your instincts and do what is best for your family. So much of it depends on your values and the ages and maturity levels of your children.
As for specific ideas... My DD has always wanted to be in the same room as me. We can do different things, but she always wants to be where she can see me and hear me. Therefore, I have always requested she choose quiet activities when I am ill. At different ages, the level of "quiet" is certainly subjective. LOL And I am sure four children are rarely truly QUIET.
So, I am talking RELATIVELY quiet.
Color Wonder activities (about the only time we used these gifts LOL)
Cards (I taught her to sort them by color and shape when she was YOUNG and she LOVED it! Number sorting came much later.)
Reading (Even before she could really read, she loved looking through a pile of books.)
Pretend nap (A game I would play with her where we'd "pretend" to nap and the ultimate result was we would really fall asleep.
Do you have any friends, neighbors, and/or relatives who would be willing to entertain a child or two for a few hours? Over the years, we've had a few neighbors who have done this for me here and there. We had no relatives within a few hours' drive for many years. My dad was 8-9 hours away and did come down a couple times when DH & I were both very sick (extenuating circumstances existed, too) when DD was really little. Now my dad lives within a two-minute walk and he'll help me whenever he is in town and home. He has a second home in another state and is very active in both places, so he isn't always available.
I hope you feel better soon!!!!