Is it possible to obtain a negative value for the variance or the standard deviation?

Standard deviation is the square root of the variance, which is the average squared deviation from the mean and as an average of some squared numbers it can't be negative.

The smallest value variance can reach is zero. This is when all the numbers in the data set are the same, therefore all the deviations from the mean are zero, all squared deviations are zero and their average is also zero.