You can make a very basic small loom out of a single piece of thick cardboard.
This is ideal for small weaving, around 10cm - 20cm size. You can use it to make small patches (or even small pouches/bags). I learned to do this at primary/elementary school, so it is very accessible.
This gives you a loom that you can use for tapestry weaving. I will explain the basics of how you would start weaving with such a loom, but details of how to weave are outside the scope of the question.
To make a simple patch: Take a piece of cardboard a little bigger (1 cm in each dimension) than the size of the patch you want to make. Cut notches 0.5 cm long, 0.5 cm apart on two opposite edges. Now you have a loom!
To set up the loom, you need to add the "warp" threads, which form the basis of the fabric. I would recommend something like knitting yarn for this. Arrange the loom so that the notches are top and bottom. Take some yarn and wind it a couple of times around the top-left notch, ending with the long end of the thread at the back. (This is just to keep the thread secure while you weave. At the end you will remove it from the cardboard and you can weave in the end of this thread.)
Bring the thread from back to front through the top-left notch. Then take it down to the bottom left notch and thread it through to the back. Bring the yarn to the front through the second notch on the bottom.
Bring it to the next notch at the top, round the back and so on... Keep going until you have filled all the notches. Wind a couple of times around the right.
Now you can take your "weft" thread and weave it through under and over starting on the lower right and going through to the lower left. You can use something like a pencil to press the threads firmly down to the bottom. Then weave back from left to right, alternating the over and under threads. Press the thread firmly down again. Keep going until you have filled the card. Then you can remove it from the loom, weave in the ends, and you have a piece of woven fabric.