Supposedly the threshold of hearing is 0 dB SPL although most rooms have quite a bit more noise than that. A dynamic range of 100+ dB is usually considered adequate though
@amirm and
@j_j use a higher number, around 130 dB worst-case, I think? Many brands reputable or not can achieve 100 dB SNR (signal-to-noise ratio), far fewer 130 dB, and in the real world it will generally be less. And there is the fact that we can can actually pull signals out (hear them) when they are below the noise floor, since noise is random but signals repeat so we can hear the signal patterns "through" the noise.
In any event your audio system is indeed a system so a single number is not all that relevant or useful. Noise adds through the chain, sort-of... It is not a linear addition, more like RSS, so total noise is the square root of the sum of the noise components squared, ntot = sqrt(n1^2 + n2^2 + ...) Assuming all the noise sources are uncorrelated (different, not related to each other). And you have to take the noise to some reference point, like the speaker output, to see where the system noise floor really is.
For example, your DAC has a noise floor, so generates some level of noise. You connect the DAC to a preamp, so the DAC's noise is multiplied by the gain of the preamp, and at the preamp's output you have the DAC noise Ndac times the gain Gpre plus the output noise from the preamp Npre. Then you go to a power amplifier that has its own gain Gamp and noise Namp. Finally you have to generate the output signal from the speakers. Assuming passive speakers, the noise the add can usually be neglected, but they do have a "gain" from input to output that converts the power amp's signal to the audio signal you hear. It is easiest to stay with the same units all through the chain so I'll use voltage (power = voltage squared / resistance).
At the amp's output you get roughly Nout = sqrt[(Ndac*Gpre*Gamp)^2 + (Npre*Gamp)^2 + Namp^2) for noise. You also have the signal from the DAC (Sdac) multiplied by all the gains so Sout = Sdac*Gpre*Gamp. Now the signal-to-noise ratio is Sout/Nout and in dB for voltages is SNR = 20*log10(Sout/Nout). That gets you the SNR of the electronics. Assuming the speakers add no noise, not unreasonable for passive speakers (and for active speakers their noise becomes the amp noise), that gets you the SNR of the system
sans room noise.
Unfortunately it gets more complicated in the real world (doesn't it always?) Some noise is more intrusive than others, and we are more sensitive to midrange noise than low or high frequency noise (see equal loudness curves). Some hiss in the upper midrange may be more intrusive than a 50/60 Hz power supply spur. If the room happens to have a peak at the MLP at 120 Hz, then noise from the power supply may be more intrusive than the hiss at a lower level. And of course there are household items that raise the noise floor, appliances, HVAC, barking dogs, crying babies, etc. If the noise in your listening room is 50 dB, then a 40 dB noise floor at the amp's output may be adequate. If you have a nice quiet room that hits 10 dB SPL noise then 100 dB of SNR through your electronics may still be audible (intrusive or not). Only studios hit that sort of number IME; my room is well-isolated, well-treated, and hits around 30 dB or so with me in it looking at (not holding! on a tripod) an SPL meter.
So the simple answer, choosing 100 dB or 130 dB or whatever, is just a starting point, alas.
HTH - Don
