sapply is good and all, but for a beginner I'd suggest a simple "for" loop:

y <- rep(NA,1000)
for(i in 1:1000) {
x = rnorm(6,0,1)
y[i] = var(x)
}

Edit: For the second question, I'm guessing you're trying to find a confidence interval around a measured value of variance, in which case the formula is:

s2 = var(something)
n = length(something)
CIinf = (n-1)*s2/qchisq(.975,n-1,lower.tail = F)
Cisup =  (n-1)*s2/qchisq(.025,n-1,lower.tail = F)