Is there a point where null hypothesis testing is superflous? Say you're writing a paper and you have the following data

Each bar represents an average over a 120 values. I want to determine if the left five and right five sets of data are significantly different from each other, e.g. left datapoint one vs right datapoint one and so forth.
The smallest difference is the fourth data point respectively, with the left being about 1.9 times bigger than the right.
Is it common to perform a null hypothesis testing for a difference this (apparently) significant?
Or, asked differently, is it conceivable to have data where you would have to accept the null hypothesis even though the data looks to be this skewed, especially over 120 samples.
 A: As a general rule, the whole point of hypothesis testing is to formalize concepts like "this looks significant" by actually giving some meaning to what "looks significant" means, so yes, you should pretty much always do hypothesis testing if your goal
is to test a hypothesis, of which testing difference between means is certainly an example. There is one valid caveat: if your goal is to only say things about the sample of data you collected and make statements about that sample, then you don't need to test, the observed differences are the true differences (of that sample). Hypothesis testing, in a imprecise way, is about generalizing the observed results to a larger population. If you had a sample of 1st graders and wanted to know if the boys or girls are older on average, you can just compare the means and no need to test anything, but if you wanted to use that data to say something about the average age difference first graders by gender in a more general setting, then you need need to test your hypothesis.
To answer your specific question, you can of course have situations where you could fail to reject the null even when you observe such a difference, and indeed, for any difference, you can easily think of data that can show that result without it existing. In particular, notice that you are looking at sample means, and you're missing the fact that you may be dealing with data that has large variance.
To illustrate concretely, here's a really simple example in R:
> set.seed(132)
> x = rnorm(120, mean = 0, sd = 100)
> y = rnorm(120, mean = 0, sd = 100)
> mean(x)
[1] 3.754229
> mean(y)
[1] 7.399619

They are both 120 draws from the same normal distribution with mean and variance, and the observed mean of y is about two times that of x! Of course, if you tested this, the sample variances would be huge, and so you'd fail to reject the null. Indeed, we have:
> t.test(x,y)

Welch Two Sample t-test

data:  x and y
t = -0.28222, df = 235.34, p-value = 0.778
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -29.09253  21.80175
sample estimates:
mean of x mean of y 
 3.754229  7.399619

We fail to reject the null at any reasonable level (pval = .78). So yes, test -- things that seem "obvious" by looking at features of the data (such as means) ignore a lot of information about the data.
