In principle all you need is a DC voltage source (e.g. battery), a particle source (e.g a heating coil powered e.g. by your wall socket AC electricity would provide an electron source), some wires, and metal plates.

You'd hook up the plates to the battery terminals using the wires, place the plates such that the negative plate is near the coil and the positive plate on the other side of the coil, and at a bit of a remove.

The issue of course is that the energy the particles would be accelerated to is very low (e.g. for a 1.5V battery, the highest energy would be 1.5 eV). Another issue is that much of the flux would be scattered by the air between the plates. Finally, there would still be the problem of measuring the particle flux to show that you actually have particles being accelerated. However that last is not part of the accelerator, just a verification.

The above is a bare-bones, primitive linear accelerator. Building a circular accelerator is much more complicated in terms of the magnetic fields needed to keep the particles in a circular "race-track" as well as having much lower likelihood of the particles staying in the beam after multiple circles through atmosphere.

You would improve both types of accelerator greatly if you could evacuate the air from the volume in which the particles travel. This would require a vacuum containment vessel and a vacuum pump.

For comparison, the energy per particle in the highest energy electron accelerator ever built (the circular Large Electron Proton machine or LEP at CERN, outside Geneva, Switzerland - www.cern.ch) was ~45 GeV. That's 30,000,000,000 times higher than the 1.5 eV mentioned above. Another big difference is that LEP was a collider, where two counter-rotating beams (electrons in one, positrons or positive electrons in the other) were steered very precisely, such that tight bunches of electrons crossed at the same time as tight bunches of positrons in very tiny interaction regions at the centers of several multi-story sized detectors.
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