I live in a very urban area and have an extremely small yard. I have experience growing in buckets and have done both aeroponics (where your roots are suspended in the air and get water at regular intervals -- usually a rail system with PVC or something similar) and hydroponics, where your roots actually in water all the time. All of my bucket and rail pants are soil-less. I have a raised bed garden outdoors for larger plants and root veggies that I mix my own soil in (about have bought and half home made compost as my soil is also not good)
I have also started experimenting/researching a small scale aquaponics system (growing plants and fish together soilessly). My goal this year was to get the fish going but work has gotten in the way and I am very behind where I wanted to be.
You can grow a ton in a small space with no soil at all -- it's really amazing. You can sort of make it as simple or as complicated as you want but the important thing to remember is that your roots are going to need 3 things basically: nutrients, water, oxygen. You can get oxygen into a hydro system through bubblers/stones or through a ventri. You can't just give water and get great growth. You can grow in a light-weight medium like hydroton or lava rocks and then add your own nutrients to the water.
If you are looking for something a little less sciency and a little more gardeny you can do large pots or raised beds with mixed soil and do a drip system or even make a wicking barrel.
There are TONS and TONS of DYI videos on youtube. It can start to feel a little overwelming but once you get started it is easier than it seems and I think it is a lot of fun to mess around with and work out your own system.
I am not sure what direction you are thinking of going in but here are some of my favorites:
vertical garden:
hydroponic rails:
wicking barrel:
(rob bob on Youtube is my HERO)