You don't need much of anything to practice Feldenkrais which is one of its many appeals for me. Two props, however, come in handy for Feldenkrais classes - a mat and some kind of head support. If you are just starting with Feldenkrais, there is absolutely no need to buy anything.
Mat (free, basic)
A yoga mat can work in a pinch, but ultimately you'll need a surface that you can slide on. The combination of a yoga mat with a blanket on top provides cushioning while also allowing the possibility of sliding body parts. This is the basic mat setup I recommend. If you make it past this basic prop description I'll provide a fuller range of options.
If you are uncomfortable lying on the floor with these basic props you could add more layers (mats, blankets). If you are still uncomfortable, I am happy to help you find a setup that works.
Head Support (free, basic)
Folded towels work fine for head support. I suggest 1-2 folded towels for lying on your back, 2-3 for lying on your side. The amount varies by the dimensions and needs of your body. I can help you refine your arrangement in class.
Feel free to stop reading at this point if you are just getting started with Feldenkrais. If you have been doing Feldenkrais for a while and you're hooked, you may be interested in a more detailed description of prop options.