me siento - two meanings?
More or less: "I sit down" is one of the meanings, but the other is not "I feel myself", but simply means "I feel", or "I find myself" (used with a similar meaning). The context will determine the meaning:
Me siento en la silla.
Me siento mal.
Sentar(se) - to sit (oneself) down
Sentir(se) - to feel (oneself)
As lazarus1907 says, you'll almost always be able to tell which meaning of "me siento" applies by using the context.
There are far more homoyms (words which we spell the same way but which mean different things) in English than in Spanish and you can almost always tell which meaning applies according to the context.
Homoyms in English:
fair (appearance) and fair (equal/reasonable) and fair (event)
- bill (dollar/money) and bill (duck's) and bill (charge slip) and Bill (name)
(More info: http://www.editingandwritingservices.com/homonyms.html)