Is it illegal to have a picnic in the middle of a roundabout?

Depends on your state laws...

The center roundabout in a road belongs to the state/county..

It is also a dangerous place for a picnic since a larger vehicle could plow throught the center killing eveyone at the picnic..

In most all states you could not have a picnic in the grassy median strip between highways,so roundabouts are most likely the same...

Go to a public park where picnics belong