In my experience, snow will get into ANYTHING. Taping seems to do a pretty good job of keeping snow out of the barrel. I've mainly used electricians tape, but I think any tape would do.
Due to polar bears, a rifle is standard equipment at Spitsbergen except within some tiny pockets of civilization and sometimes not even there. (Though nowadays, people are encouraged NOT to enter the bank with a rifle over the shoulder).
The common practice of dressing with face covering scooter masks did not exactly minimize what some bank employees considered a problem...
Those rifles receive very little care and a lot of abuse, but the electricians tape seems to do a very good job of keeping snow out of the barrel.
I have also thought about wrapping the mechanism in some cloth with velcro closure (quick release), but I never got that far while living there. At some occations, the entire locking mechanism has been covered with snow and ice, but brute force solved that problem. Bolt action rifles are pretty sturdy stuff. But I still think it is a good idea to cover up a rifle that spends hours and hours exposed to blowing snow.