Buying your freedom was possible, but it would most likely occur only under very unique and specific circumstances.
With regard to how the slave could buy his/her freedom, specifically with what he could but his freedom, a slave could posess
peculium which could consist of money or other things of value. However, the slave only possesed
peculium by the grace of the master. Technically the
peculium belonged to the master, so if he was unwilling to allow the slave to have
peculium, then there would be no chance at buying your freedom.
It was also required that there be an agreement between the master and slave that the slave could buy his/her freedeom with the
peculium. This would most likely only occur when the
peculium outvalued the worth of the slave, thereby being more profitable for the master to "sell" the slave his/her freedom than to keep him/her.
It is quite unlkikely that a slave could "nickel and dime" his/her way to freedom by squirreling away a
sestertius here and there. The most likely avenue for gathering enough
peculium to buy your freedom would be by slaves who acted as an agent for their master through trade or other profitable means and the slave being alowed by the master to keep some of the profits. When enough was attained and the master agreed to it the slave could buy his/her freedom.
One question that came up while I was looking into this was if allowing the slave to possess
peculium was by the grace of the master and the slave could only buy his/her freedom if the master agreed, then, why would the master agree to any of this since by law the
peculium belonged to him and he did not have to sell the slave his/her freedom?
My best guess would be that if a master allowed a slave to keep a portion of the profits as his own and that one day he/she might buy her fredom that the slave would work harder and hence make more money for the master then just merely "punching in and punching out" the timeclock (please forgive the moden analogies!)
I hope that sheds a little light. If you want to look into it further check out
(The) Roman Servus, an article in William Smith's
Dictionary of Greek and Roman Antiquities. It explains
peculium in much more detail as well as many other aspects of slavery in Ancient Rome (with cites to ancient sources as well).
-Severus