If atoms are 99.99% empty, why do we feel objects?



You feel them because if you push on solids, they push back. Molecules are made of atoms, and comprise a set of nuclei at fixed distances, and electrons moving around in accord with 
the Schrödinger equation. Now you may feel that the electrons are tiny, and most of the atom is space, and it is if you fire alpha particles at an object, as Rutherford did, but recall 

those alpha particles are very energetic and are small nuclei without electrons, so you can think of them as extremely high-powered bullets, not to be bothered by a few electrons.
Now the electrons about molecules give an electric field, and recall that these electrons are moving extraordinarily fast compared with your finger, and as far as you are concerned, the 
electrons create a stationary electric field of intensity corresponding to an electrostatic field that would arise from a stationary electric distribution proportional to the probability of finding 

an electron at each point. The target object has this stationary field, but so does your finger, and the two electric fields repel, because the electron – electron distance is smaller than the electron – nucleus distance. 
This is known as the Hellmann Feynman theorem, and you can see the proof at: Feynman, R. P. 1939. Phys. Rev. 56: 340-43. This paper is well worth reading because (a) it gives a lot 
of physical insight into why solids are, well, solid, and (b) it explains the origin of the van der Waals forces, which arise because while the electron – electron repulsion predominates, at first there is a mutual polarisation that causes the electrons to move closer to their own nuclei, and this leads to a small overall energy well.
The quantum-theoretical answer to the question is that atoms are not 99.99% empty. In spite of what many descriptions in popular texts, and even in high school physics or chemistry text books, try to convey, they are, ultimately, misleading metaphors.

Yes, an electron does take up a finite, near-point-like amount of space, as does the nucleus, but only if you measure it. If you leave it alone, the electron is everywhere around the nucleus. An atom is filled with smeared out electrons, which don’t occupy a particular near-point-like amount of space, but, instead, exist as probability clouds, which are, in turn, genuine physical objects, called atomic orbitals.
Wolfgang Pauli then formulated his Pauli exclusion principle, stating that two or more identical fermions (such as electrons, neutrons, protons, quarks) cannot occupy the same quantum state within a quantum system simultaneously. In short, the likes of Ehrenfest, Dyson, and Lenard, showed that this principle, in tandem with the longer range Coulombic force (to which previous answers here refer), is responsible for the everyday macroscopic observation that two solid objects cannot be in the same place at the same time.
So, why do we feel objects? 1: their atoms are not 99.99% empty in their natural state. 2: the atoms in our fingertips are pushed back by the atoms of the object.

Quite interesting: you don’t really ever touch the atoms of an object. There is always the tiniest gap between your skin and the object. The repulsive effect is that strong at those short distances! (But enough to fire up the nerve endings underneath our skin which makes it feel like we touch the object.)

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