Most GPS receivers today have internal data logging. The amount of storage required for a particular session depends on several things: the length of the session, the number of satellites above the horizon, the epoch interval, and so forth. For example, presuming the amount of data received from a single GPS satellite is ~100 bytes per epoch, a typical twelve channel dual-frequency receiver observing 6 satellites and using a 1-second epoch interval over the course of a 1-hour session would require ~2MB of storage capacity for that session.
The miniaturization of storage continues. The cassettes, floppy disks, and drives used with the Macrometer are past, and extraordinary amounts of data can be stored in small convenient devices.