how can i access a set of objects that are saved in another set of objects

//This code show you how to make set of obecjts , so just do this for a3,a4 as well.
Set<Abilities> a1 = new HashSet<Abilities>();

a1.add(new Abilities("sing"));
a1.add(new Abilities("dance"));
a1.add(new Abilities("shoot"));

  Set<Abilities> a2 = new HashSet<Abilities>();


 a2.add(new Abilities("sing"));
 a2.add(new Abilities("dance"));
 a2.add(new Abilities("shoot"));


//You should remember to change from array of abilities to set in the definition of your Person class.
Person p1 = new Person(a1);
Person p2 = new Person(a2);

Set<Person> persons = new HashSet<Person>();
persons.add(p1);
persons.add(p2);

we will use iterator with set here.

       Person temp = null; 
Iterator<Person> iterator =  persons.iterator();

while(iterator.hasNext(){
  Person tempPerson = iterator.next();
for(Person p : persons){ //This for loop will compare with all other objects
   if(p!=tempPerson){
  if(compareSets(tempPerson.getAbilities(),p.getAbilities())){
  throw new SameAbilitiesException("Persons with same abilities");
     }
   }  
  }
}

The Method which you can use to check if two sets are the same.

//check whether set2 contains all elements of set1
private static boolean compareSets(Set<Abilities> set1, Set<Abilities> set2) {
    if(set1.isEmpty() || set2.isEmpty()) {
        return false;
    }       
    return set2.containsAll(set1);
}

if you want to keep your code with Array :

 Person temp = null; 
    
    for(int i = 0; i < Person.length; i++) 
          {
    // Here it will be array of type Abilites not Person
              Abilities[] tmp = persons[i].getAbilities();
//This for loop will start from first index to compare
             for(int j=0; j<Person.length;j++){
if(i!=j){ // This if because we don't want to compare same array with it's self 

       if (Arrays.equals(Person[i].getAbilities(), tmp))
                   throw new SameAbilitiesException("Persons with same abilities exception!");
}
            }
    }