You get this achievement by having your dogs "paralyze" the bears IMMEDIATELY after they drop onto your area. I found out that you don't get the achievement if at least one bear still starts roaming your farm: this happens when there are more bears dropping at the same time than you have dogs. Even if you first cage and store a paralyzed bear, and the dog that is now available starts to paralyze the "roaming" bear, and you capture all of them, you still don't get the achievement.
Because of all this, the widely suggested level to get this achievement, Pork Street 2, which is the first level with dogs, didn't work for me (however, shout out to
Fire Hawk D, who did manage to find a way, check the comments for more info). You start with three dogs, but the bears drop in groups of four. So I could not get that to work for me, because there would always be a "roaming" bear.
Much later in the game, on the level Product Street 6 (which is just after Crossroads 6), you'll get this achievement by naturally playing the game. You start with only two dogs, but the bears will only drop in pairs anyway. After they are paralyzed by the dogs, and thus can't wreak havoc on your farm, just cage them, and then either move them to your storage, or let them run away. Finish the level, and since all the bears were caught after being immediately paralyzed by your duo of dogs, you'll get the achievement.
A word of warning though: after the three minute mark (= after the gold time has passed) a trio of bears will drop. Since you want to get golds (for all the achievements) anyway, this shouldn't be a problem, because you should've finished the level by that time. If it is a problem for you to stay under three minutes, however, there is a solution: you start this level with loads and loads of cash, so simply buy an extra dog and you should be alright.
Edit: Should you experience any trouble unlocking this achievement using my solution, check the comments for more ways to unlock it, on either Cake Street 5 (sell extra animals at the start, buy 2 extra dogs (credit to
Cheese Empanada)) or you might get it naturally as well on Meat Street 3 (credit to
Mr Granstaff)