This plant is a few months old. I'm in zone 10A. Although the vines continue to spread, it has always looked sickly like this with curled up leaves.
If this is not a vertical picture, which I don't think it is, allow the melons to grow up a trellis. This is will help in the natural growth of your melons. Then make sure to water your melons on a regular basis, providing the soil enough water to stay moist. You don't want to overwater the melon but your vines look rather dry. So I would recommend you increase your water.
I'm not sure how far apart you grew your melons but typical space between melons should be about 36-42 inches apart.