Digital goes straight down to DC with no issues. The only limit on the low end is that most devices aren't DC coupled, for various electrical reasons, and roll off on the extreme low end. However it is generally VERY low. A Benchmark ADC-1 goes down to 2 Hz and if you want, you can find DC coupled ADCs. Now try and find a microphone that can pick that up, and better yet, try and find an analogue tape that goes that low.
Digital is king of bass and that's part of the reason we've seen an increase in the use of heavy bass in music, is that digital allows for it. You couldn't do it very well with analogue, and not at all with many of the systems (like phonograph).
In terms of high frequencies, well you can record those with digital if you want, we just generally don't bother because there is no point. You can't hear it, you can't feel it, etc. Really, this is something that is pretty easy to test, and indeed has been tested. Our ears don't do that high, we can't hear it.
However it CAN be a bad thing, as what you can get is aliasing. Your gear may not be able to handle those high frequencies properly and your speakers or amp may generate aliased subharmonics that you can hear. So you end up getting distortion, sound that wasn't there in the recording that your setup is playing because it can't handle the ultrasonic content. Hence band limiting can be quite useful.
So digital sound is generally recorded at 48kHz (or 44.1kHz) because there is no reason to waste space recording the higher frequency content, which there is very little of anyhow (look at a spectrograph of it some time if you are interested), cannot be heard, and can cause problems on lower end gear, and even some high end gear.
Problems with recording and reproduction vs reality do not have to do with the digital storage medium, rather they have to do with the analogue recording and playback components (mics and speakers) as well as dealing with placement issues. If you want something that sounds quite real, get yourself some binaural recordings, and some good headphones to listen to them with.