Definitely can be done.
"Few tips, snow is a great insulator, but if it starts to melt and you get wet you will be in dire straights. Staying dry is the name of the game, even sweating too much is dangerous. make sure to account for ventilation. Also dig a "cold well" in your cave, this lets any cold air sit at the bottom, while your up top with the warmer air. a door is key, but you need airflow too."
- some survivalist blog
Just look up survivalist snow camping on google
But yeah, just some basic research and this can totally be done. But remember, the bigger the snow bank the bigger the room. Also put a hole above where you'll have your fire. Don't want it collapsing in the middle of the night haha.
"The ski bum trades security for face shots, the future for the moment. Considering how hollow the promise of a corporate career has become, who can say the ski bum is not the wiser investor in his or her youth?".
"I feel the same way about disco as I do about herpes." - Hunter S. Thompson