I've been living in various apartments and enjoying grassroot motorsports for nigh on a decade now. A few thoughts...
- One underrated perk of living in an apartment downtown is that you may be able to walk, bike or take transit for regular trips, and thus you may not need a car that is commute-friendly. You are freed from the drudgery of commuting and get to choose a car based on the fun stuff you want to do!
- Limited space does kind of discourage having a car that is too much of a project. For a long time I had only an E30 or only a Miata, and while they were both fantastic, the need for upkeep on a 30-year-old BMW or the desire to modify the Miata and supplement it with a larger more practical car were too great. Versatility + reliability are the name of the game. I have ended up with a BRZ, though it pretty much let me down immediately on the reliability front...
- If you don't have a huge commute, high-performance tires are fine for regular use. RS4s especially so. Had a set of Continental ECS on the E30 that were great for cross-country drives, extremely fun for autocross (if not super competitive) and pretty okay for trackdays.
- Apartments with separate storage areas are a godsend. These should be more common in larger buildings. I have a lock-up area downstairs that can accommodate several sets of tires. Conversely, smaller buildings with independent landlords may be more flexible in terms of creative storage arrangements.
- Local venues may offer storage of varying quality/cost. A trackside garage is a bit rich for my blood but they didn't seem to mind a few ice-racers mouldering in a nearby field...
- Wrenching becomes a social occasion by necessity when you can't do it at home! You'll make great friends with people who have tools & garages.
- Consider limited space an opportunity to practice tidiness in your gear inventory. I've got everything neatly packed into trackday/autoX box that lives in my trunk but can easily be brought inside and stuck in a closet if need be. Currently it's a big Rubbermaid bin though the kit in hunter47's earlier post would be a nice upgrade.
- Duckzero mentioned a Fit. I have also put some thought into these considering the popularity of the Sundae Cup at Gridlife events. Depending on your location they may be a good choice! A Miata obviously is fantastic, but also depends on having a relatable group you can run with. I found that Miata-oriented groups tend to do wheel-to-wheel whereas HPDE events tend to have varieties of cars that are on the whole dramatically faster than a stock-ish Miata. If these are not issues for you, Miata may indeed be the answer.