"..should I apply now?" No.
"Or should I continue on my path until I have (closer to) the necessary experience?" Yes.
"Or both?" No.
"...would applying to a company for a position and being rejected, hurt your chances with that same company a few years down the road?" It could, though probably not likely. It depends if they keep a database of applicants, like I did at two employers, and if the two resumes differed inexplicably.
I would suggest one of two courses of action (in order of preference):
1) Learn all you can about the advertised position, as well as, the Co. via the Co. website, LinkedIn, Facebook, etc. Search for online industry newsletters that mention the Co. Try to learn who had the position, who filled the vacancy, who the hiring manager is, and etc. Contact the hiring manager with an general inquiry letter expressing your interest in working for the Co. and mention any relevant info. you learned in your research but make no mention of the current vacancy (as if you didn't know of it).
2) Or, do all the above, and contact the Co. with the acknowledgement you don't have the prerequisites for the current vacancy but would like to apply for any position that might lead to it.