Even if Bertrand is not charging Albert for the storage of his boxes, a legal relationship of bailor and bailee still arises between them. This is because bailment for safekeeping does not require payment - it can be gratuitous (although it does not have to be).
Gratuitous bailment for safekeeping can arise in different circumstances, for instance using a free coatroom at a restaurant or theatre.
Because a gratuitous bailment benefits only the bailor and not the bailee, the obligations which arise are minimal. The bailee must take reasonable care of the goods, and typically the goods must be returned upon the bailor's demand.
One common situation of bailment for safekeeping is when a seller of goods still possesses them prior to delivery, although ownership has already passed to the buyer. But in this situation the seller is treated as a bailee for reward, so they have slightly higher responsibilities than in a gratuitous bailment.
Note: If the parties had agreed that the bailee would be paid for keeping the bailor's property safe, there would be a higher duty of care on the bailee. The next example illustrates an instance of bailment for reward.