Rainbow lorikeets squawking and kookaburras laughing here!
To the OP, how much are you cranking things up (either at recording or on playback) to hear the noise?
As has been said, all electronic equipment has self noise to a greater or lesser extent. However, in any decent gear this should be all but inaudible at levels in normal use. Similarly, any room (except for an anechoic chamber) will have a certain degree of room ambience--which, again, is normally well buried in the signals you're actually recording but which can be audible if you record silence and crank up the levels during recording any/or playback.
My suggestion to you would be to set your recording controls to the sort of level you'd typically use for recording.
Once you have a minute or so, use your DAW and zoom in on the vertical scale just to see what sort of level the noise is burbling along at. If the noise (combined room and equipment self noise) is lower than about -80dB, then you have nothing to worry about at all. If the noise is between -70 and -80dB it's beginning to become audible but is likely to be buried in the recording. More noise than that and, at normal recording levels in the -18 to -12dB range, it'll likely become audible as levels are brought up in the mix.
Fixing it? Well, the only true way to fix it would be better sound proofing and acoustic treatment and better gear (with less self noise). A gate can turn off the sound whenever to source material falls below a certain level but the noise will still be there whenever the gate is open. Personally, rather than an automatic gate, I prefer to use fader automation or envelopes to take down the level of every track during periods of silence. Yeah, it's work but the results can sound better than a gate (at least to my ears).
Or, if you DAW has noise reduction, grab a sample of pure noise and run the reduction algorithm. Don't try to eliminate all noise; just reduce the level so it's not objectionable and you should avoid too many artefacts on your music. Also, doing 2 or 3 light passes can often sound better than trying one big reduction.