Well, let's check some obvious stuff (just in case). You're loading the camera and developing the paper in darkness or safelight suitable for the paper, correct (I never know what level people are working at when they're making oatmeal box pinhole cameras)? If you develop a sheet of the same paper straight out of the storage box, it should be completely white. Given this, put a sheet in your camera, put it out in the sun, and leave the pinhole covered. Let it sit there for five minutes or so, then take it into the darkroom and develop the sheet. If that sheet isn't also perfectly white, your camera has light leaks, and the solid black paper is just fogged from the time spent in the camera.
If that paper isn't fogged, then your exposure is mongo too long, which suggests your pinhole is much too large. For a standard oatmeal box, the hole should be pretty close to .018" or about 0.45 mm, which would give around f/300; assuming ISO equivalent of 6 for the paper (few papers are actually this fast), "Sunny 16" exposures should run between about 60 and 90 seconds (without correcting for reciprocity failure -- paper should be okay in this range), but if your hole is 1 mm (about .040"), you'd get a correctly exposed (if unnecessarily fuzzy) negative at about 20 seconds, and if the hole is 2 mm it would only require about 5 seconds. To turn completely black, though, the paper would have to be overexposed by much more than the two stops you'd get with 20 seconds and a 2 mm hole, so I'd guess your oatmeal box isn't anything like as light tight as you think.
Another quick test: with the hole closed, put a bright flashlight inside the box and close it, then look at it in your darkroom to check for leaks. Do it again with the flashlight the other direction, of course. If you see light anywhere other than the imaging pinhole, you need to seal up the leak.