First do prime factorization of $2014 - 2^{1} \times 19^{1} \times 53^{1}$
Now to get a factor of $2014,$ we can choose any combination of the prime factors including $0.$ i.e; $2^{0}$ and $2^{1}$ are possible and similarly for other prime factors also, there are $2$ possibilities. So, the total number of positive integral factors
$= 2 \times 2 \times 2 = 8$
(When all the powers of prime factors are $0,$ we get $1$ and when all the powers are maximum, we get the given number.)