I am by no means an expert on this, this answer is ~20 minutes of me googling things, but I would say that in theory it is probably possible to do this with an arduino, at least at the lower transfer rates anyway, and certainly from a "will it kersplode my board" point of view - probably not since ethernet runs at +/- 2v (ish) and ~10mA (ish). However:
The ethernet protocol is exceedingly complex, with many layers needed to actually communicate with anything useful, all of which you would have to implement from scratch (there won't be a library to just drop in) and I'm not completely convinced that an entirely software-based implementation of the stack would fit on the arduino (but I could definitely be wrong there - and actually, on further thought, this probably is wrong, but I reckon I couldn't write it so it fit, but I'm a reasonably average human being).
Arduino is all about hacking and re-inventing wheels just for the fun of it but I am absolutely certain that there is no fun to be had re-engineering Ethernet and TCP/IP and all of the stuff in betweeen from scratch (obviously my opinion, I enjoy all sorts of things others don't, so YMMV).
My suggestion would be - if you don't like ethernet shields, look into getting a 'raw' ethernet controller chip (like the one soldered onto an ethernet shield) and build it directly into whatever device you're trying to make first, that'll give you plenty of exposure to how ethernet works at that level, then you can decide if you're still up for re-inventing the brain that invented the wheel. Good luck!