In one word: No.
Why Standard Deviation Can’t Be Negative
First, let’s stop for a moment and think about what standard deviation represents.
It measures variability in a data set.
When you have some set of numbers and calculate its standard deviation, the resulting number tells you to what extent the individual numbers in the set are different from each other. If all are about the same (like 252, 249, 253, 251, 254), standard deviation will be relatively small. If there are big differences (like 252, 11, 840, 305, 64, 5846), standard deviation will be much bigger.
What if all the numbers in the data set are exactly the same (like 252, 252, 252, 252, 252, 252)? Then standard deviation will be exactly zero.
Can you get an even smaller standard deviation (which would have to be negative)? No. You can’t have a data set which is less diverse than one where all numbers are the same, right?
To conclude, the smallest possible value standard deviation can reach is zero. As soon as you have at least two numbers in the data set which are not exactly equal to one another, standard deviation has to be greater than zero – positive. Under no circumstances can standard deviation be negative.
Why Standard Deviation Can’t Be Negative Mathematically
The above was the common sense explanation. There is also a mathematical explanation, based on the way standard deviation is calculated.
Standard deviation is the square root of variance, which is the average squared deviation from the mean and as such (average of some squared numbers) it can’t be negative. See more detailed explanation here: Can variance be negative?