There are many options.

First let's estimate the memory capacity you need. For legible voice you should take around 8k 8-bit samples per second, so for 20*30=600 seconds it is 600*8kB=4.8MBytes. That's not impossible but also quite a lot.
There are compression techniques, most of them lossy, for reducing this requirement while retaining legibility. The most sophisticated enable something around 1:10 compression (they are quite computationally intensive, probably more than an average '51 is capable, but let's not consider that for the moment). That would bring down the memory requirement to around half a megabyte.

Now consider technology. You can choose (S)DRAM (cheap for a couple of MBytes but complicated access/interface/refresh; medium consumption), SRAM (easy interface, more expensive, low consumption on backup), EEPROM/FLASH chips (more expensive, nonvolatile, slower write), FLASH cards (cheap in high capacities, nonvolatile, slower write).

Now consider interface - parallel (fast, but requires many signals/pins), serial (slower but requires only a few signals/pins, some derivatives have direct support).

Jan Waclawek