The context is comparing the loudness changes in two sounds.
An eighth note at quarter equals 120 is 0.25 seconds long
So is it talking about four eighth notes beamed together, or is it something to do with intervals?
I assume "quarter 120" is an odd way of saying the tempo/beat is 120.
A tempo of 60 means one beat/quarter per second. A tempo of 120 means two beats or quarters per second, so that makes it .5 seconds per quarter.
An eight = half of a quarter = .25 seconds
Loudness has nothing to do with this.
You’re not parsing your sentence right. You should understand “An eighth note at [quarter equals 120] is 0.25 seconds long.”
“A quarter = 120” is a tempo indication meaning “you should play 120 quarters per minute”, i.e. a quarter last half a second. An eighth note thus lasts half that, that is 1/4 second.