The spider web filling stitch is another stitch from my 1964 Coats & Clarks booklet One Hundred Embroidery Stitches. It's also shown in another of my embroidery books but done with ribbon instead of thread and it resembles a rose much more than a spider web. So feel free to experiment with the material you weave through—it doesn't have to be the same as the base threads, which will be completely covered by the time you finish.
Start by drawing a small circle on your fabric—mine is about the size of a quarter. Mark the center and five evenly-spaced points around the outside edge. Bring your thread up in the center.
Take five straight stitches from the center out to each of the five points that you marked. Bring your thread up again in the center in between two of the "spokes".
Without going through the fabric, weave the thread under and over the five straight stitches, making sure to alternate between spokes.
Because you're working with an uneven number of spokes, the next time you come around the circle you'll be doing the opposite of what you did before, so each spoke will alternate under and over. As you go around, guide the thread around the previous one with your thumb. You can keep the thread tight near the center but make it looser as you work outward. Otherwise, if you pull too tightly, it will pull over the previous thread. This is a little hard to explain but you'll understand once you start working this stitch.
When you want to stop, take your thread to the back slightly under the edge of the outer thread and near one of the spokes. This will help to hide the end.
Here's what it looks like when you stop the woven threads partway out and keep the ends of the spokes showing. (I've used a fade-away marker for my circle, so the lines will disappear in a day or so).
And here's another version with a second color in the center (just stop one color and start another) and the weaving taken all the way to the edge. The weaving creates a raised-off-the-surface texture and would be a cool way to do flowers!