In a large, deep skillet, heat the olive oil. Add the sliced onions, cover and cook over moderately low heat, stirring occasionally, until they are very soft, about 10 minutes. Add the minced garlic and cook, stirring, until fragrant, about 3 minutes. Add the sliced red cabbage, cover and cook, stirring occasionally, until the cabbage is tender, about 20 minutes.
Meanwhile, in a large pot of boiling salted water, cook the linguine until it is al dente. Drain the pasta well, reserving 1 cup of the pasta cooking water. Return the pasta to the pot.
Scrape the cabbage over the pasta. Add the reserved pasta cooking water and toss well. Season with salt and pepper. Transfer to bowls, top with the feta and serve.