Is it possible to...

Is it possible to make a Recording Module with an arduino? Could you record your voice for 10-30 seconds without filling the memory, or would you need some seperate memory, like an sd card.

You would need external memory of some kind. At Atmel only has 1k or 2k of RAM.

Short answer is no there is nowhere near enough memory on an Arduino. How much do you need? I'll leave that for someone with audio-grabbing experience, but you'll probably need an SD card shield or something similar.

Thanks for your replies. I don't know exactly how much memory the sound would take up. If I have a 1gb sd card connected to my arduino, it could fit many many sound files, but will it even work? Can the arduino itself write and read from an sd memory card and have the ability to play back the sound that was recorded?

Telephony-grade audio has a bandwidth of 3kHz.
Assume no coding, you're going to need 6000 samples per second (Thanks to Mr Nyquist).
Even if they're only low-quality 8 bit samples, that's still nearly 6kbytes per second.
You may have some spare processing capacity for compression/decompression, though generally, the higher the compression, the more processing you need to do.

Do you need to record and playback on the same device?

You can use the Voice shield: