I used a T-shirt for the cover and sewed shut the bottom, neck and one sleeve while shirt was inside-out.
After turning the shirt right-side-out, I stuffed the other old clothes in through the open sleeve. I left the sleeve open so I could stuff more things in later - old towels, ripped clothing, etc. She's got quite a fluffy little bed now.
Pillows can also be made this way. Just use a pillow case as the cover, stuff to desired firmness with desired materials, and sew shut.