If each neuron in the brain could only help store a single memory, running out of space would be a problem. You might have only a few gigabytes of storage space, similar to space in an iPod or a USB flash drive. Yet neurons combine so that each one helps with many memories at a time, exponentially increasing the brain's memory storage capacity to something closer to around 2.5 petabytes (or a million gigabytes).
For comparison, if your brain worked like a digital video recorder in a television, 2.5 petabytes would be enough to hold three million hours of TV shows. You would have to leave the TV running continuously for more than 300 years to use up all that storage.