Actually if you're sick and breastfeeding you want the baby close so you can nurse MORE so they'll get the antibodies and hopefully WON'T get sick. Now in a perfect world there will be another healthy parent around to do all of the childcare chores EXCEPT nursing and will leave you alone to wallow in your misery except for when the baby needs fed. In my REAL single mom world the one time I was sick with a fairly young baby, I moved everything we NEEDED with in arms reach of the bed, nursed a lot and onl got up to use the potty/empty the barf bucket.
When the kidlet is sick we still cosleep, I just put most of the towels we own under the kidlet then when he gets sick I only have to take the dirty ones off the pile....
ETA:
Plus my son was a HUGE comfort nurser whenever he didn't feel well-- he was pretty much glued to the boob when sick so there would have been NO WAY to put him somewhere else when he was sick. The good news was I NEVER had to worry about dehydration until he was almost 4!