I want to convert a section of lawn, 3 feet wide and 11 feet long, in to a garden. The question is, how do I get rid of the grass, without using a tiller or herbicides?
You can dig up the sod or rent a sod cutter.
Here is a link that might be useful: Sod busting
If you are not in a hurry to plant right away, cover the entire area with a thick mulch of newspapers and then cover the papers with good garden soil or compost. If you do it now, you should have a good soil without grass or weeds by late winter or early spring.
You can get used carpet and lay on the area you want for the garden and by spring it may me grass free. I cut strips of used carpet and put in paths between growing beds to keep out weeds. You could till the soil up and form raised beds and sort of bend the carpet up on edges to hold the soil.