Say for instance a person who was strong enough to lift double his body weight. If he placed his hands under his bottom and tried to lift$^1$ himself$^2$ off the ground, could he?
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$^1$In a slowly, controlled motion; sudden jumps etc. are excluded.
$^2$ All parts of himself without touching or being supported by the ground in any way.
Best Answer
No, it's not possible.
Why? Think about, for a second, what you're trying to achieve. The most important concept that you're missing is that a force is an interaction between two objects. One applies a force to the other and the other applies that same force back. You might be wondering how anything gets done/moved then?
Well, if you think about the simplest mathematical expression of force $F= ma$, it shows that the force exerted is the product of mass and acceleration. When we say that an object exerts the same force on an object, it does. But the gist of it is that objects differ in mass. Sometimes drastically.
$$m_1a_1 = m_2a_2$$ $$F_{1\rightarrow2} = F_{2\rightarrow1}$$
Here's that in written form. Object 1 applies a force to Object 2. An interaction. Object 2 applies a force to Object 1. Forces are equal, yet, they move. Why? Think about it, if object 1 has $m_1 = 100m_2$ (100 times the mass of object 2) and exerts a force of, let's say, 300 N on it, it will move it without breaking a sweat. Object 2 will apply that same force on it, but because it has such low mass, it won't make even a dent in the acceleration of Object 1. That's a crucial part of that interaction. Here, even the math agrees:
$$100m_2a_1 = m_2a_2$$ $$100a_1 = a_2 $$
The acceleration of object 2 is, linearly, 100 times bigger more than object 1. Now, let's get to your problem. Can someone lift himself? You push yourself and yourself pushes you. Since we're talking about the same object, this reduces down to two same masses exerting the same force on each other. And the net force is $0$.
Same goes for cartoonish ideas like moving a boat by blowing into the sails. This is a repercussion of the translational symmetry of our Universe, conservation laws and common sense™. Hopefully, this is simple enough to make you see why it doesn't "work".