No. A computer doesn't need speakers to beep. My roomate's worked fine with no speakers.
And, your roommate listened to what on his computer? Music, audio content from the web, or just beeps? While it's true some computers have internal speakers, the big majority do not. So, they require external speakers.
However, external speakers are usually not needed for the beeps that computers emit during startup or to indicate certain hardware errors. So, if that's all you care about hearing, then you probably don't need speakers.
Oh, I just saw Cityscape's post. I see I'm duplicating his comments to some extent.