Here's a simple Monte Carlo simulation. Assuming that the population distribution is Beta with parameters 80 and 1.1, I draw samples of 5 observations, then plot the histogram of their mean.
title 'Population distribution'
title 'Sample mean'
You can see that the sample mean does not fit neither normal nor t-distribution, so t-statistics would be useless. I don't think that you can derive the distribution of the sample mean without knowing something about the distribution of the population beyond its bounds. Your sample variance is 26, i.e. too high relative to the size of the population and its bounds in order to apply CLT here or other asymptotic analysis reliably.
Now if instead of assuming some random population distribution, let's fit Beta distribution to your data.
% fit Beta distribution to the sample
% generate many samples of 5 observations
title 'Population PDF'
title 'Sample mean distribution'
pd = makedist('tLocationScale','mu',phat(1),'sigma',phat(2),'nu',phat(3))
disp 'Conf. intervals of population mean'
This gives an output:
t Location-Scale distribution
mu = 88.2223
sigma = 1.82489
nu = 89.9698
Conf. intervals of population mean
The 95% confidence interval seems reasonable, but the t Distribution fit is not so good, it's not skewed as the sample mean distribution. You could do something along this line, so long that you have a good idea about the population distribution. I still don't tink that the confidence interval would be reliable given the tiny size of the sample and its high variance.