Yes, you can build your very own nuclear fusion reactor in your house! But first, a few warnings:

-This project includes lethal voltage levels. Make sure you know your high voltage safety or have a qualified electrical advisor.
-Potentially hazardous levels of x-rays will be produced. Lead shielding of viewports is a must!
-Deuterium, an explosive gas, will be used. Make sure to check for fuel leaks.
-All the other inherent dangers of a home engineering project of this degree (a wide gamut of potential injuries, damage to the checking account, and the loss of general sanity)

Here are the minimum required materials:

-A vacuum chamber, preferably in a spherical shape
-A roughing vacuum pump capable of reaching at least 75 microns vacuum
-A secondary high vacuum pump, either a turbo pump or oil diffusion pump
-A high voltage supply, preferably capable of at least 40kv 10ma - Must be negative polarity
-A high voltage divider probe for use with a digital multimeter
-A thermocouple or baratron (of appropriate scale) vacuum gauge
-A neutron radiation detector, either a proportional He-3 or BF3 tube with counting instrumentation, or a bubble dosimeter

-A Geiger counter, preferably a scintillator type, for x-ray detection and safety
-Deuterium gas (can be purchased as a gas or extracted from D2O through electrolysis - it is much easier and more effective to use compressed gas)
-A large ballast resistor in the range of 50-100k and at least a foot long
-A camera and TV display for viewing the inside of the reactor
-Lead to shield the camera viewport
-General engineering tools, a machine shop if at all possible (although 90% of mine was built with nothing but a dremel and cordless drill, the only thing you really can't build without a shop is scratch building the vacuum chamber)

A quality high vacuum chamber is required for the fusor to operate. Sometimes an appropriate chamber can be found on eBay, but generally it is best to make one. Parts can be scrounged for several hundred dollars, or purchased new for $500+.

Get two stainless steel hemispheres, purchase two corresponding conflat-flanges (8" flanges in my case), bore out holes for accessory flanges, and then TIG weld it all together. Flanges are typically either of the KF or the conflat style. Conflat can be seen in the image below as the flanges with bolts, and KF (kwik-flange) are seen as those with only clamps holding an o-ring on the mating surface. Only weld on the inside, never on the outside (since virtual leaks can be formed if both inside and outside are welded). If you've never TIG welded before, it would be wise to have someone with experience do it as the welds must be flawless with no pin-sized holes or porous areas to hold a vacuum.

After machining, thoroughly clean the chamber and avoid getting fingerprints in it since these will outgas, which means at vacuum pressure molecules in the oil of finger prints or machining oil will become vapor and make it hard to maintain plasma stability or reach a good ultimate vacuum level.

Install the oil diffusion pump (or turbo pump if you have a bit of luck scrounging or a higher budget). Fill the pump with quality diffusion pump oil to whatever fill level the pump documentation suggests, attach the inlet to a valve which then connects to the chamber (see diagram), and attach the outlet to a mechanical backing pump capable of reaching at least around 75 microns (any higher and the diffusion pump will not operate properly or the oil will oxidize quickly).

Make sure the pump is sufficiently cooled, many oil diffusion pumps require water cooling, smaller ones such as the one pictured can get by with a decent air flow.

Once this is assembled, turn on the mechanical pump and wait for the vacuum to reach at least 75 microns. Next you can test the high vacuum pump by turning on the boiler on the diffusion pump. After it warms up (could take a while), the vacuum should rapidly drop below the single micron range.