Is it possible to jump by using only my arms? We can see that basketball players use their arms to jump higher. 
Is it possible to lose contact with the ground by only moving my arms?
I know that the calculations have something to do with momentum, but how massive and how fast should the arms be moved to achieve this effect?
EDIT:
The given answers consider moving only the arms in a downward motion to force the rest of the body to go the opposite direction.
I've found that there is another way which requires some kind of contact with the ground.
Let's say we have 2 heavy dumbbells. We then proceed to rapidly throw the up in the air, without letting go. I think that if we throw them hard enough, the body will be forced to go along.
If we remove the dumbbells I guess it might be possible as well, but would require bigger speed change.
Are the equations required to calculate this similar to moving the arms only downwards?
 A: There is the principle and the practice.
Let's look at the principle. If you have two arms, each with mass $m$, and length $\ell$, we could restate your question as saying: "how, and how fast, do I have to move such arms to make my body lift off?". That's quite easy.
Assume you are holding your arms out sideways. Their center of mass is at $\ell/2$, and their total mass is $2m$. If you move your arms down rapidly, there will be a net vertical force on the rest of your body. Move the arms fast enough, and you will lift off:

Simple conservation of momentum says that if you move your hands down at velocity $v$, you are moving a mass of $2m$ down at velocity $v/2$ and so the net downward momentum is $mv$. In the absence of gravity, this would be compensated by the body moving up at velocity $V = \frac{m}{M}v$. However, gravity does play a role - which is why you have to move your arms fast if you want to get liftoff in this way.
To figure this out we have to remember that $F\Delta t = m\Delta v$ - the faster the change in momentum, the greater the force. In this case, we need to overcome the force on arms plus body: $F_g = (2m+M)g$. If we consider that you accelerate the arms from zero to $v$ in $\delta t$, then the velocity $v$ can be calculated from
$$(2m+M)g \delta t < m v\\
v > \frac{2m+M}{m} g \delta t$$
Putting in reasonable values for mass of arms 6 kg each, and rest of body 60 kg (lean athlete), you need $v/\delta t > 60 m/s^2$. It might not be humanly possible to get sufficient force from the shoulders to move the arms that quickly - although it's not hard to see that the force will be similar to the force needed to perform "the cross" - a very demanding gymnastics move (here demonstrated by Yuri van Gelder doing the L cross - requiring incredible effort from shoulders and abdomen):

The reasoning that this is the same force that would provide the lift needed for the body - but where the move as shown above is done statically, you would have to do this dynamically - so there would be not just force, but force and speed.
So while it is theoretically possible, I doubt it could be done practically.
Of course moving the arms in various ways can certainly help - I am traying to answer the question "can you lift off with just your arms". I know I couldn't...
A: Theoretically, you can try to move your arms downwards (for a short time), that would tend to move the rest of your body upwards. The mass of the arms is approximately 10% of the total weight of the man's body (http://www.timesdaily.com/archives/weighing-in-on-individual-body-parts/article_4729f5a7-c039-5649-910e-ee18a03435e0.html ). To have the rest of your body to accelerate with acceleration g (to counteract gravity), your arms should move with (average) acceleration 9g. Is it possible with human anatomy (note that the shoulders are static) - I don't know.
