Firstly, you have to make a few decisions... Arcade collecting is not a cheap hobby and does indeed take up a lot of space. On the other hand, you don't have to buy EVERY game you've ever played when you were a kid, you could just stick to a couple.... Or one, if you think you won't 'bite' the arcade bug!
There is another route here... But its not one that particularly interests me. PC's have come such a long way in such a short amount of time that arcade games are virtually 'pre-historic' by comparison. It is now possible to play more or less ANY arcade game using an arcade emulator (MAME). Better still, its even possible to build a cabinet to house an old PC to run MAME to have that 'arcade' feeling! More information can be found by searching the internet using any search engine - try Google.
That's a route that will probably appeal to most people, but this site is about REAL collecting!! Arcade board (PCB's - Printed Circuit Boards) are freely bought and sold all over the world every day. Mostly, trading takes place on Ebay UK or Ebay US. But don't just sign up and start over-bidding... Most collectors will tell you that they all got ripped off at one time or another, do some research!! Ebay has great search tools for completed items and there are plenty of boards to go round for everyone. If you lose out on something, don't fret!! Its never long before another one appears for sale! Obviously though, as the old collector saying goes "its only worth as much as someone is willing to pay for it" which is why you must NEVER post a 'how much is XXXXX game worth' on forum sites (UKVAC) because that's exactly the answer you'll get!
Right! You now know where to buy arcade boards but what do you do when you've got one?? How do you play it? The best option here is to buy a cabinet, either a full upright or maybe a cocktail one. Surprisingly, an upright doesn't take up that much space - its 'footprint' is only a little larger than the monitor that sits in it, although they tend to be a little on the tall side! Saying that though, there are a variety of cabs on offer and it is possible to buy a mini-cab/cabaret, shop around on the links page. The other type of cabinet is a cocktail, you may remember these in pubs or youth clubs. The older ones have joysticks either side and players take it in turns (the screen flips/inverts for the second player). Newer ones are pretty rare but as you can see from the photo link both players can play on the same side of the table (simultaneously if the game allows). Now while these cabs seem like a good idea (and indeed a good way to smuggle an arcade machine past the wife!) they do have their drawbacks. They do tend to be uncomfortable as you're playing looking down on the screen with the joystick jutting out towards you, also you may have trouble finding one that has a rotateable monitor (some games are horizontal, some vertical) but if you want a fancy coffee table its a bit of a talking point!
Lastly, theres the supergun/MAK its just an arcade console that plugs onto the board and plays through a standard television. It breaks my heart whenever I see people buying them for £150+ theres really not much to them. ALL the hard work is done on the arcade board itself, all a supergun does is route the power and take care of the sound/video output/player controls. Honestly, for around £50 worth of parts you can build yourself a supergun - it's so easy!!!