A couple of comments firstly:
1) Skirmish games are very popular right now, and most seem to require 2'x2' to 4'x4' playing areas. Whatever you choose to play though, you will still also need some extra space or a small side-table for things like rulebooks, dead models, etc.
2) Some skirmish games have very high terrain requirements, which means that despite the small playing area, you will still need reasonable storage for the terrain. There are workarounds to this though!
3) Consider what scale of game you would like to play. 15mm is very cheap, but 28/30mm is more popular due to the prevalence of "big name" games.
Next up, I don't really know what your preference is for - making terrain, modelling, painting, or playing. Nor do I know if you are more likely to play narrative or competitive games, or if you would prefer a solo game or one that you can play at a local club. I will therefore make a variety of suggestions!
Space games can be played without much terrain, and with reasonably few models that you don't have to paint (or spend much time painting). Two good examples are the very popular X-Wing, and the newly-released Dropfleet Commander. A further advantage is that if you use something like a terrain mat, it can just be rolled up and put away when you're not playing. If you want to spend more time on the modelling and painting side, these games still offer good opportunities. If you fancy X-Wing, I believe the rules are free to download from Fantasy Flight's website (although you will still need cards, templates, special dice, and tokens to play).
Malifaux is a very fun game, played on a 3'x3' table, and is reasonably popular. All your old models are still valid and playable, although you will need one or more new decks of model stat cards. There is a mini rulebook for cheap, although the full rules are free from Wyrd's website too. It's pretty competitive, but also has loads of special narrative scenarios and terrain setting rules - so you can normally adapt whatever terrain you have well enough within reason, or even play on a custom board representing something like the inside of a theatre or a saloon, or underground in mining tunnels, or whatever.
Warmachine and Hordes are again very popular, and again have full free rules online, and are played on a 4' x 4' table. Terrain favoured by most players tends to be more abstract and "playable" than fully scenic, and some 6-8 mixed elements are usually enough to keep you going for a while. The new edition of the game (Mk3), has a heavier focus on the magical generals and their associated beats/robots this time around, so although there are lots of units available for each faction, you can get away with only a couple to keep model numbers down in your collection. It's a very clean game rules-wise (like Malifaux), and is strictly about gameplay first and foremost. This can make it feel a bit cheesy if you're not expecting it, but is actually great fun if you are that way inclined.
Infinity is another popular skirmish game, but has high terrain requirements despite being played on a 4' x 4' board. All the rules, model rules, and army-building tool are fully free online, available directly from Corvus Belli. Infinity is rather complicated to learn, and has a lot of rules to remember, which results in a fairly steep learning curve. Until you have a good grasp of the game and rules, it will not feel especially balanced - although perseverance is worthwhile as the game is very action-oriented and fun (and balanced!).
GW made a number of classic skirmish games in the past, and these remain surprisingly fun and popular today. Rules for Necromunda (sci-fi gang warfare) and Mordheim (fantasy gang warfare) are freely available online (Yaktribe has all the original digital files available for download that GW released freely before they dropped support for the games). These games are not always especially balanced, but they are characterful skirmish campaign games with low model count requirements. Like Infinity though, they both need lots of terrain to be properly playable. Even though the models for both games are out of print, there are still loads and loads of alternative options (including converting figures). You will need a 4' x 4' space to play these.
There are also many (many!) more skirmish games out there that you might like, but I cannot speak for their wider popularity. If you have anything specific in mind that you'd like to try, please ask as I'm sure that we can guide you towards some suitable rule sets!
If you are willing to try 15mm wargaming, you at least have the realistic opportunity to collect several different factions so that you can offer opponents "loaner" forces. For some brilliant inspiration on skirmish wargaming at this scale (with space constraints and on a budget!), I suggest looking through
Tiny Solitary Soldiers blog.
Finally, there are a number of boardgames, boardgame/wargame hybrids and "wargame lite" style games that may interest you. Imperial Assault and Descent (rules free for both on Fantasy Flight's website), Deadzone 2nd Editon, and the recent GW boardgames (Deathwatch Overkill, Betrayal at Calth, Prospero Burns, and Silver Tower) are all good examples, but there are loads more. With the GW games I mentioned, you can normally buy the complete game without miniatures in otherwise mint condition for about £15 on ebay (and then buy whatever models you like to use with them).