That is the way it must be in a free society. How much a person saves is up to him. He may perceive his probability of unemployment as negligible and therefore save little for that reason, although he may save a lot for other reasons. The benefits of saving are heavily taxed unless you are saving for an approved purpose and go about it in an approved way. If people want to set up separate accounts for retirement, medical expenses, college costs, and so forth, they certainly may do so, but there is no reason why they should have to go to the extra bother.
Money is fungible.
On the other hand, he may perceive that he faces a strong probability of unemployment and, therefore, decide to set aside a fairly substantial portion of his income. His employer may facilitate his saving preferences with some kind of program, but if not, it is quite easy for an individual to establish a savings plan with any number of financial institutions. If he becomes unemployed, he would draw upon the funds in his savings account at whatever rate he thought best. Notice that reliance upon individual initiative desocializes the risk of unemployment. No longer would one person's misfortune or laziness be a burden to everyone else.