Why can't two hydrogen atoms be slowly "pressed together" (fused to form helium) with some advanced device (not collided at high speed)? I apologize for my lack of physics knowledge, and I won't be gaining much any time soon, since I'm focusing mostly on work and physical exercise.  I can't help but wonder about this: 
Questions:
What are the main barriers preventing physicists from slowly pressing two hydrogen atoms together to fuse them into helium?  (as opposed to very many high-speed collisions)
Is such a device that would do that physically impossible to construct, or just very much harder to construct than one of those magnetic-confinement, toroidal, plasma trap, fusion reactors?
I won't ask any more ignorant questions for a while (maybe about 6 months). 
 A: It takes a lot of force to fuse hydrogen.  Nuclear fusion requires the atoms be brought close together, causing them to experience very high electrostatic repulsive forces.  Only once you get them close enough together do you see the quantum effects which draw the nuclei together into fusion.  The atoms have to have enough energy to push past this barrier, known as the Coulomb barrier, before  they can engage in fusion.  In chemistry, the analogue of this is the "activation energy" required for a reaction to occur.  For a typical deuterium/tritium reaction, this energy barrier is 0.1MeV.
If you "pushed" two atoms together, you would have to provide all of the force required to overcome the electrostatic forces.  This is quite a lot of force, because the electrostatic forces increase proportional to the square of the distance between the nuclei, and they have to get very close.  This is also a very unstable repulsive system, so the nuclei would like to escape.  It's like trying to press two billiard balls into each other.
Instead, it is much easier to accelerate the hydrogen atoms over a very long time period to a high speed, and let kinetic energy do the hard part of overcoming the Coulomb barrier.
A: Each hydrogen atom (a lot of hydrogen is found in molecular form, rather than as a single atom), is a proton with a paired electron . They are attracted to each other because of opposite electrical charges. 

If you start with a gas, and you compress it, you ( depending on the particular element), can end up with a liquid. In fact, liquid  hydrogen is the propellent which many rocket engines use.
Imagine putting an arbitrary number  hydrogen atoms in a chamber, with a piston on top, that you can use to compress the atoms together. As you squeeze more and more, the hydrogen  atoms or molecules will move faster and faster, as they will repel each other because the electrons in each has the same charge. 
This mutual repulsion is the same reason you don't fall through the floor in your house, because of the electrons resistance to being forced together. So it's a very strong force, especially as the atoms get closer. 
"Some advanced device" is still going to be based around getting the atoms together and it is very difficult  to think of anything that would overcome the electrostatic force keeping the atoms apart. Not only do the electrons repel each other because they have the same negative charge, but the protons in the nuclus of each atom will also repel each other because they both have the same positive charge.
They are two sets of massive springs basically, which  get increasingly stronger the more pressure you put on them.
A: Here is the binding energy curve that plots the energy released by fusion for the low mass elements

To get $H^2$ ( a deuteron) i.e the fusion of two hydrogen nuclei, enough energy must be given to strip the electrons from the protons, and then to overcome the electrostatic barrier of the two $+$ charged protons, so that the attractive strong force takes over, and then the weak interaction turns one proton into a neutron a positron and a neutrino, and fusion into a deuteron becomes  a reality.

The fusing of two protons which is the first step of the proton-proton cycle created great problems for early theorists (of the sun) because they recognized that the interior temperature of the sun (some 14 million Kelvins) would not provide nearly enough energy to overcome the coulomb barrier of electric repulsion between two protons.
With the development of quantum mechanics, it was realized that on this scale the protons must be considered to have wave properties and that there was the possibility of tunneling through the coulomb barrier.

You ask:

what are the main barriers preventing physicists from slowly pressing two hydrogen atoms together to fuse them into helium? (as opposed to very many high-speed collisions)

One cannot pick an atom and press it. One can strip a hydrogen atom and make a proton beam and collide protons on protons. This is very inefficient in energy production  way to generate deuterons, a lot of energy is lost in accelerating beam particles that will never collide, because collisions are a statistical effect and depend also on the crossection of interaction. See the answer here for details. .
The  use of plasma to generate fusion, mimicking the sun, gives an efficient method to use the statistics of scattering within the plasma to get a positive energy output from the process of proton proton scattering. In fact , if you look at the curve more energy is gained by using other atomic combinations.
