Oh it can be done, mountaineers have been doing it for decades.
Your going to need:
a -40 bag, or a -20 and wear your warm clothes inside of it.
A sleeping pad is also mandatory, possibly two. One closed cell foam and an air mat on top of it.
A serious 4 season tent. Not some cheap walmart piece of shit. Something that will vent moisture well is absolutley fucking mandatory. Your shit getting wetter and wetter over a period of a few days is going to fuck you. It should also be able to put up with a proper blizzard. So good materials, proper additional guy lines midway up the tent. Could you do a bivy? yeah, for a little bit, but thats really only a backup shelter in those conditions, the condensation problem is going to be unavoidable in a bivy. It takes careful disciplines management to not fuck up with a bivy in those extremes, and its not something that was ever really intended for long periods of time.
And of coarse all the other little things, hats, face coverings, gloves/mitts, multiple layers, thick jackets ect. ect. Quality heavy duty boots. You might also find things like a snowshovel indispensible for snow camping.
Now the problem with all of this is you can go two ways with the type of gear
1: Cheaper, HEAVY, BULKY, possible not very durable. Synthetic sleeping bags, older cheaper tents. Heavy thick ass wool layers. Your pack will probably weigh 70lbs MINIMUM.
2: EXPENSIVE, difficult to find, light, packable, durable.
By expensive I mean thousands of dollars to equip yourself. This is light, packable downbags, high quality 4 season lightweight mountaineering tents. If you spent a year prowling every thrift store, outdoor consignment shop, and online deal you could maybe pull off 1400$.
This is absolutley possible, I’ve done it, I’ve met others who’ve done it. It’s just going to take resources, effort, and knowledge. If you make too many mistakes in summer in 40 degrees you’ll probably just have a bad time. In -40 you’ll be dead.