The other answers haven't completely addressed these two points, which I'll do here:
Cave Johnson says that the repulsion gel does not interact well with the human skeleton, but Chell gets covered in it and appears to be fine.
Cave Johnson is a funny guy, he uses lots of euphemisms that show his callous disregard for human safety in his experiments. Here's a list of his quotes. The actual quotes about repulsion gel and skeleton are here:
"Oh, in case you got covered in that repulsion gel, here's some advice the lab boys gave me: DO NOT get covered in the repulsion gel."
"We haven't entirely nailed down what element it is yet, but I'll tell you this: It's a lively one, and it does NOT like the human skeleton."
If you interpret that literally, you might think there's some dangerous reaction when the gel comes in contact with the human skeleton, or something along those lines, but Cave Johnson is actually talking only about being covered in it. Also, knowing how he talks, the "not liking the human skeleton" part is a euphemism for you will bounce uncontrollably and break all your bones! This can be demonstrated in the game if you cover objects like cubes in the gel - even from a resting position, they start bouncing up uncontrollably. From the wiki:
Objects such as Storage Cubes and Turrets can be covered in Repulsion Gel, which causes them to bounce around erratically.
For Chell this is no problem most likely for gameplay reasons.
By the way, here is something Cave Johnson says before all this:
"You're not part of the control group, by the way. You get the gel. Last poor son of a gun got blue paint. Hahaha. All joking aside, that did happen - broke every bone in his legs. Tragic. But informative. Or so I'm told."
The reference to breaking leg bones does two things: it reassures the player that she is safe because she has that Advanced Knee Replacement, and also it tells you what Cave Johnson is getting at with the "skeleton" part.
The Moon Rock gel is supposed to be extremely toxic since it appears to be killing Cave Johnson. Again Chell is covered in it and is fine.
Apart from the short duration of the game which others have mentioned, one theory is that Cave Johnson is dying from silicosis via inhalation of moon rock dust:
"The bean counters told me we literally could not afford to buy seven dollars worth of moon rocks, much less seventy million. Bought 'em anyway. Ground 'em up, mixed em into a gel."
"And guess what? Ground up moon rocks are pure poison. I am deathly ill."
From here, two assumptions can be made:
- Only moon dust is dangerous, and not moon gel. Chell only comes in contact with the gel form.
- If we don't take Cave Johnson literally (an easy thing to do) when he says "pure poison", then silicosis is a likely explanation since in real life, moon rocks are just regular rocks. And again, silicosis requires inhalation of dust, which isn't present with the gel.