Question: I've been "swimming" once in a while for years, while on vacation for example, and never had a problem. Recently I've been learning the crawl (freestyle). I've finally got myself up to 20 meters or so without running out of breath. But now, after swimming the crawl for a while, I feel sick. I come out of the water and lie down, and things spin (as if I've drunk too much), and I feel a little like throwing up.

I'm pretty active. I run and things like that. Is the nausea a sign that I'm doing something wrong, or is it just part of getting acclimated to this stroke?

Answer: 
This is most likely either a breathing problem, or a fatigue problem. If your muscles aren't tired at all then it's probably breathing. Given your description it sounds like you are hyperventilating. If your muscles feel like jelly then it means that you're pushing too hard. I've seen plenty of people puke (and have done so myself) after going harder than they were ready for.

Either of these is rather dangerous as breathing can make you pass out, and pushing too hard can make you cramp. Both lead to drowning.
If you can't do more than 20 meters your stroke is really bad and/or your breathing technique is bad. Bad swimming technique is incredibly exhausting, about the same a drowning. As a beginner you will need to breath on every stroke in order to get enough air. And as people have said here, you need to get all the CO2 out before the next breath, not easy to do at first. Try swimming with fins until you get the hang of the stroke. You won't struggle to stay afloat and the additional speed will make it easier to breathe.