Since you're talking specifically about vocal harmonies, sounds like, I'll just talk about that, (though most of the concepts apply to instrumental harmony as well).
First off, the biggest factor in how harmonies will sound is the voices you have to work with. While the traditional ensemble includes soprano, alto, tenor and bass, that is by no means a rule, and any variation is fair game. I must take issue with some previous comments here that something should be voices a fifth here or a third there. There are no such rules. You'll get the best sound by trusting your ear.
The actual note writing for a barbershop is not very different from that of any choral writing, but what makes that barbershop sound is more about the vocal style being sung, the a cappella-ness of it, the fact that it's four men, and the kinds of songs typically sung. Crosby Still & Nash's sound is definitely not barbershop, but it's three men, so why not? Because of everything just stated. What makes CSN sound so good is how in sync they are. Listen to "Helplessly Hoping", how every vowel is voiced exactly the same, every inflection in volume matches, and their cutoffs are in perfect sync. They also were lucky in that their voices just had a nice blend. Peter Paul and Mary were also good for those reasons.
Contrast that with something like the BG vocs for the Carpenters, smooth as a baby's cheek, which had to do with how they were recorded and the meticulous nature of the blend. Likewise, Mancini's harmonies for a song like Charade is distinctive because he's got four male parts against two or even only one female part, which gives it that rich deep warmth.
As for writing them, first become familiar with the vocal ranges of your singers, and try to stay in the ranges that are most comfortable for them. If any one part gets too low or high, you'll have balance and tone problems. Know the chords ahead of time of course, and then it's a relatively simple matter of making sure every chord is spelled out with at least one voice. Where it gets sticky, and where the real art comes into it, is in the considerations of the HORIZONTAL aspect. Harmonies will sound much better if each part sounds like a real melody that would be OK if sung on its own. Basically, avoid big leaps. Even though our ears may not be able to pick out those melodies when everyone sings together, the fact that they're in there makes a much better sound. Spend a lot of time on that. Also, try to prevent voice crossings (where one voice is higher than another, then their lines "cross" so that that voice is now lower), because that's awkward to hear and also to perform. It's like doing Sudoku sometimes, because you need to make sure all the chords are represented, each voice sounds like a melody, and everyone is singing in their range. But it's not really that hard.
The above paragraph is in cases where the voices are doing all the harmonic heavy lifting, but in many cases the vocal harmonies are mostly for color and aren't holding up the whole harmonic structure. In that case you have a lot more freedom, and are probably writing fewer parts, but the same rules apply. Make each line its own melody, try to prevent voice crossing, make them easy to sing.
HTH!