First, painting any floor brown will darken the whole room (or cost you more for the 150 watt bulbs and the energy to light them). Very dreary.
Second, there may be some paint that sticks to linoleum, but on a floor my bet is that it will not last long.
You said that there is red oak underneath? Rip that linoleum out, sand it, and finish it! There is a significant time investment, but the materials cost is low and you get a spectacular result.
If you rip everything up and find that the underlayment is waferboard or pine surfaced plywood, I would invest in laminate (more expensive than you want) or if the underlayment is in good condition go retro with vinyl composition tile (costs less than a buck a square foot if you install it).
But I think that paint over a surface that is designed to allow everything to be spilled on it to be washed off may be a short-lived solution.