Can electromagnetic fields be used to deconstruct and reconstruct atoms? I was thinking one day and came up with a theory after reading about how scientists were studying anti-matter by using electro magnetic fields to separate matter from the anti-matter they made.  It got me thinking would it be possible to use very powerful electromagnetic fields to break down the atomic structure of objects or build things in this way?
Is this atomic reconstruction with electromagnetic fields theoretically possible?
That is, is it theoretically possible to use electromagnetic fields generated by a machine to separate the parts of an atom thereby deconstructing an object on the atomic/subatomic level?
I'm not asking about breaking molecular bonds but rather actual atoms apart.  If it's possible to break atoms apart with electromagnetic fields, is it also possible to use a similar process to assemble them?
 A: It is doable, easily, if you consider time-dependent EM fields - or, more specifically, pulses of light. This is because light can be focused into a tight focal spot of ~ wavelength size, thus squeezing a lot of energy (and thus EM field) into a tight spot. This is impossible using static fields, which can't reach maxima except at their sources. Just as importantly, one can prepare really short pulses of radiation that compress into a short (i.e. few femtoseconds) time the energy stored in a laser medium over a millisecond or so. This allows quite modest energies - of millijoule order - to create electric fields comparable with the atomic electric fields (as in John Rennie's answer) for long enough to ionize just about any atom.
Note that this does not need a macroscopic sample (although actually observing stuff usually does!). If you manage by some means to get a single atom to sit still at a known position (which is a doable but nontrivial exercise) then firing enough strong laser pulses will ionize it, releasing one electron. You can of course release more electrons but it will naturally get harder and harder (so that triple ionization is the most you can hope for). Photon energies (which are proportional to the frequency) are not really a concern in this high-intensity regime, where multiphoton ionization and tunnel ionization take over as the main mechanisms as one goes from optical to IR and lower frequencies.

Regarding the converse in your question, we have nothing like the technology to "atomically print" a table, atom by atom, and I would question the feasibility of such a scheme. Optical tweezers, for example, can be used to move molecules around and depositing them on a surface could potentially work, but their spatial resolution is limited (by the wavelength, which can't be too short or stuff starts to break). Even worse, you'd have to work at it for something like geological-scale times to get anything sizable.
On the other hand, metamaterials are a good example where we can build materials to atomic-scale detail so that they have the macroscopic properties we want, including such exotica as negative refraction indices or strongly chiral media. While it does not exactly fit the bill you ask for, I think it is fundamentally more cool with the added bonus of being feasible.
A: It's exceedingly difficult to tear apart an isolated atom with an electric field. It's a lot easier to ionise a gas of many atoms, but the mechanism is different. Let's stick to an isolated atom for now to see why it's hard.
Consider an isolated hydrogen atom. The binding energy of the proton and electron is 13.6eV and the average separation is about 0.05nm. So to tear the electron and proton apart you need a field gradient of about 13.6V per 0.05nm. Converting this to more sensible units gives about 2.7 $\times$ 10$^{11}$ volts/metre, and this is far above what we can create in the lab.
I did say it's easier to ionise a gas of many atoms. Suppose you apply a strong electric field to a hydrogen gas, and suppose there is a stray electron somewhere in the gas. Let's not worry where the electron came from; maybe a passing cosmic ray produced it. Anyhow, the free electron will start to accelerate along the electric field lines and at some point it will crash into an atom. If the collision energy is significantly greater than 13.6eV the electron will ionise the atom it hits, and now you have two free electrons. Those two electrons will in turn accelerate and collide with other atoms and you get an electron avalanche. The mean free path of an electron in air is about 0.5 microns, so the corresponding electric field is 27MV/m and this is easily achievable. There's more on this in the Wikipedia article on Paschen's law.
A: Electric strength =5.145175226 X 10^11N/C
Magnetic strength  = 235125.6635 T
so Velocity of the electron =
 E/B =2.188266.117 m.s^-1 = C/137
