Well, let's start with the minimum size of an object. The formula for escape velocity is:
$V_{escape} = \sqrt{2 G M/R}$
Okay, let's say that you don't want to accidentally jump off. Let's say that you can do that with about a 3 m/s jump. Let's give a density of $3 g/cm^3$, and let's just say a uniform sphere. Putting all of that gives:
$3=\sqrt{2*3000*G*(4/3*\pi*r^2)}=8000*G*\pi*r^2$
Solving for r gives about 2.3 km. So that's your absolute lower limit. Of course, you might want something that you could walk more upright. Let's say you wanted at least 50 m/s for an escape velocity, as that you would come down relatively quickly. That would come to about 38 km.
There is no object with a solid surface that is too massive to walk on, so that's okay. It must have a solid surface, so that takes out the gas giants. Aside from that, the only objects to be wary of are those with a toxic atmosphere, which would be Venus, and Titan. Venus would be difficult with any conceivable technology. Titan might be doable, but it would require some special work, and likely wouldn't work for long. The main problem with Titan is the atmosphere is flammable, which could cause some interesting complications if there was any kind of a leak.
Next, let's look at heat and cold. According to Wikipedia, a space suit can operate between -156 °C (−249 °F) to 121 °C (250 °F). Saturn is about -178, so you'd have to be inside of Saturn's orbit. The upper temperature range would be reached for an object at Venus's orbit, probably not that far.
Very last thing is extra hazards. The main hazard would be radiation, which only really affects Jupiter's closest moons, Io and Europa. The further ones should be okay.
Okay, that leaves hundreds of asteroids and NEOs, the moons of Mars, Mars, the Moon, Earth, and the further large moons of Jupiter, and the trojan asteroids of Jupiter, and probably some other stuff I'm missing. With some work, we could probably get to Mercury and the outer solar system. Venus, and the gas giants, well, not likely. Titan might be tricky, as it's atmosphere is literally flammable.