It is fairly easy. You have to extend the Min and Max Life in Basic attributes to the length of your clip (or at least to the part where snow would be accumulated).
The Physics attribute of the Snow effect is already set to Stay. Nothing to change here.
Then convert your ground to terrain, or make an artificial Dummy ground from a box and enable physics for it (Rigid Body - Static)
The only limit here is a Snow effect Quota. Max value is 100000. Meaning it only can have that amount of particles accumulated.
When that is depleted, effect would die out (but snow would stay to the duration of lifetime). With Emit Rate and Velocity I have set, it has depleted in 35 seconds
But I guess you may have another one start to emit (I did not try that).
As to everything else like a fence, again you need to either encapsulate props into Rigid Body Dummies or enable Rigid Body for props themselves.