Explaining Infinite Sets and The Fault in Our Stars

In watching The Fault in Our Stars I could not help but cringe at a line that flew in the face of mathematics and subsequently ruined the movie for me:

"There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities." - John Green

While walking out of the theater I tried to explain to my friends why there were, in fact, exactly the same amount of numbers between 0 and 1 as 0 and 2, but Cantor and bijective functions are not great learning tools to English majors.

Does anybody have an eloquent or elegant way to enumerate this phenomenon using an example accessible to those not familiar with advanced mathematics?

Look, don't worry about it. The author is absolutely correct if by "bigger" he means bigger Lebesgue measure rather than bigger cardinality. Cardinality is just one way to abstract our intuitions about size and it isn't obviously the best one to use in all situations (especially in this kind of situation where it returns highly counterintuitive results).

Assume Alice has a basket with balls in it, one for each real number between $0$ and $1$, which is written on the ball. OK, it is hard to imagine so many balls - or even how one would manage to write down an arbitrary real number on such a ball, but that is not the point here.

Bob also has such a basket, also with one ball for each real number between $0$ and $1$. If there is any concept to make sense of this at all, we can only say that Alice and Bob have the same number of balls in the basket.

What if Bob took out his marker pen and painted a green dot on each of his balls? Of course, the two buddies still have the same number of balls.

What if Bob instead would prepend the symbols "$2\times$" before the number written on the ball? Of course, the two buddies still have the same number of balls. The difference to the previous example is minor - green dot or a digit and a times symbol, that does not make a difference.

What if Bob now for each of his balls replaces the number ($x$, say) written on it with the number $2x$? Of ocurse they still have the same number of balls. The difference to the provious example is minor - whether the ball has "$2\times0.314159$" written on it or "$0.628318$" doesn't matter.

Now on closer look, Bob notices that in his basket he has one ball for each real number between $0$ and $2$.

We conclude that there are just as many numbers between $0$ and $1$ as there are between $0$ and $2$.