How can I cover a 4" space under my door that lets in a draft?

Your local hardware store or Lowe's/Home Depot have door sweeps that are easy to install :-)

A door sweep should help!

https://www.amazon.com/Stopper-Bottom-Basement-Exterior-Interior/dp/B07V2QFF2Q

For an added layer of protection, you can always make a draft dodger.

https://m.wikihow.com/Make-a-Draft-Dodger

https://www.bobvila.com/slideshow/draft-dodgers-8-diy-creations-to-keep-out-the-cold-46522

You could add a kick plate to the door on both sides and then a sweep on the bottom of that. For stability, you could add a couple of cuts of 2x4s.

Ann...You can purchase weatherstripping that should work or make a draft snake that will cover the opening where the air comes in. They can be filled with rice, beans, cotton, etc.,and can slide under the door with a snake on each side. Google and see what you might like. Good luck

All of the above ideas will work to some degree. My question, What part of the country do you live in? If you live in a cold climate, while those ideas make it look better most won't do much for keeping out the cold with a gap that large. I would first measure exactly the gap at the bottom of the door. Then take the door off, measure the exact length that you need to fill. Go to the nearest lumber shop & have them cut a piece to fit. Making into two pieces and screwing them on one at a time will also work, put a good exterior wood glue between each addition, and you should have a solid addition to your door. You would want to sand, use filler to fill any cracks and after that dries sand again. Paint and you won'e be able to see the addition at all and you can add a kick plate if desired. I would also do as one other reader suggested: get a door sweep to put on to stop any draft. The rubber threshold would work best for keeping out the cold. Another option for accomplishing this, is if you have a handy friend, Ask for Help and if the door is a heavy one, you might need help to handle it anyway.

4" is a huge gap IMHO, if you live in a cold area, I would visit Habitat for Humanity Restore and buy a wonderful old door to fit to your space. Take your measurements with you and a tape measure. I got brand new mahogany doors for $99 but they have some wonderful doors for as cheap as $25.

Go to a salvage yard or a clearance center and get a door that fits. 4 inches is a lot of space, and no matter where you live, it will let in cold or hot air not to mention critters. Our builders clearance sell beautiful doors for under $100.

Go to home depot and get a strip for on the bottom for drafts it screws on and you could adjust the hight

pool noodle, cut length to fit, the slice length wise down middle, then slide onto bottom of door