Among the options provided, the football club that is not a legitimate football club is Option 4: Monte Carlo. Monte Carlo is actually a district in the city-state of Monaco, known for its luxurious casinos, hotels, and the prestigious Monaco Grand Prix Formula One race. It is not associated with any professional football club.

Let`s examine the other options to provide further reasoning:

Option 1: Arsenal - Arsenal is a well-known football club based in North London, England. Founded in 1886, Arsenal Football Club, also known as "The Gunners," has a rich history and is one of the most successful clubs in English football. They have won numerous domestic league titles and cups, as well as European honors, including the prestigious UEFA Champions League.

Option 2: Aston Villa - Aston Villa is a professional football club based in Birmingham, England. Established in 1874, Aston Villa has a long-standing history and has competed in the top tiers of English football. They have won multiple league titles and have a strong fan base.

Option 3: Chelsea - Chelsea Football Club is another prominent English football club based in London. Founded in 1905, they have become one of the most successful clubs in recent years, winning multiple domestic league titles, FA Cups, and European trophies, including the UEFA Champions League.

In summary, Monte Carlo is not a football club. It is a district in Monaco known for its casinos and Formula One race, whereas Arsenal, Aston Villa, and Chelsea are all established and respected football clubs with rich histories in English football.