I built two mass spectrometers back in the 1980s. The first was primarily a photoelectron spectrometer but was also used as a decent linear time of flight (ToF) mass spectrometer. It was built on a fairly low budget, but it was in a university with access to a lot of surplus equipment. The ionization source was an expensive UV laser but ToF MS can also be done with pulsed acceleration voltages on grids for electron impact (EI) ionization. The detector was a stack of two microchannel plates (rejects from Varian that failed image tests, we just needed a time resolved total signal). Data acquisition was with a "waveform digitizer", essentially an early headless digital oscilloscope.
The second mass spectrometer I built was in industry with a 1985 US$700,000 budget just for parts and components which included a 150mm bore 7 Tesla superconducting magnet and an ultra-high vacuum system with three deferentially pumped chambers. Obviously not a garage project, however the electronics and computing could be done at a much, much, lower cost today. In fact I have much of what is needed sitting behind me right now.
The "front-end" was a RF Quadrupole and ion source that were purchased as a module from Extrel (which is still in the business).
I was going to respond to ikrase's electrometer thread since I initially used a Keithley 614 to measure the ion current that reached the detection cell inside the magnet. No electron multiplication was needed, a beam stopping plate was just connected to the electrometer. Of course sensitivity would have been limited and we usually ran the RFQ front end as a high pass filter (not passing some set threshold of low mass ions, but everything above that mass) so the current could have been from ions of a wide mass range.
Later I switched to a Keithley 617 electrometer to use the GPIB interface as part of an ion optics auto-tune system (there were multiple ion lenses in the injection path).
If I were to build a mass spectrometer in my garage I would probably look for a surplus RF quadrupole system rather than build that part from scratch, but maybe it's not that hard. The quadrupole elements can probably be made at any machine shop that does precision center-less grinding. A pulsed ToF MS would also be considered.
I'll also note that ion trap mass spectrometers, like the Orbitrap, are hot these days. Could be a DIY possibility, likely requiring some precision machining.
ikrase: What mass range are you interested in? Will you need high sensitivity? Will there be a lot of masses in a single spectrum?
Here are a couple of resources that might be of interest (inspiration more than as a supplier due to cost). I used both of them and once lived a few miles from one (but we both moved).
http://www.kimballphysics.com http://www.sisweb.com