You've helpfully posted 30 values in the first instance. They aren't named so I've called them $y$ and I'm presuming that they are the response or outcome you want to explain. We can look at its so-called marginal distribution, but the bigger deal for the $t$ tests and ANOVA you have in mind would be the conditional distributions of $y$ for each distinct value of any predictors or covariates.
A normal probability plot (normal quantile-quantile plot) should look more like this than the plot first posted (but now removed).

There are different conventions (1) for which variable goes on which axis and (2) on whether the expected values are presented using the observed mean and standard deviation (SD), or estimates thereof, or in standard normal terms (with mean 0 and SD 1).
Setting those small points aside, the main detail is to confirm your reports to the effect that the distribution is slightly skew. As the values cover a fairly small relative range (about 35/25) no (standard) transformation will make much difference to that skewness, as over a short interval such transformations are close to linear in the observed values. It is shown above that a logarithmic transformation reduces the skewness slightly; in practice I wouldn't bother unless there were independent scientific or substantive grounds for working on logarithmic scale.
It's an assertion based on experience that the degree of normality is too slight to worry about much and I would go ahead with $t$ tests and ANOVA. But I would rather have data for your other variables to be able to check that conclusions here are robust to minor non-normality.
EDIT: The fuller data are now revealed as two groups of 30 on the same outcome.
Given yet more anonymity, I am calling the group identifier x
with values of 0 and 1, although nothing depends on any such names or values.
A superficial analysis suggests that the groups are so close in their means (29.007 versus 29.843) that any kind of test is virtually redundant.
But the extra 30 values have a mild outlier that inclines me to using logarithmic scale to dampen its effects a little, and quantile plots hint at structure that is more than (not even!) a shift of distributions:


Collectively, group 1 is lower at lower values and high at higher values. The small difference in means arises because these effects almost cancel. (The pairing here, it must be emphasised, is quantile to quantile, not part of the observation design.)
All logarithms here are natural.
Mild protest: You posted 30 extra values and then deleted them! Why do that? No one can use them in answers if you do that. Here are all the data as I have them:
x y
0 27.8
0 26.4
0 25.6
0 28.8
0 26.1
0 30.2
0 31.6
0 31.4
0 27.9
0 29.3
0 31.1
0 27.3
0 26.3
0 26.8
0 35.1
0 27.8
0 26.5
0 27.2
0 31.4
0 31.9
0 28.2
0 27.2
0 29.4
0 29.5
0 30.2
0 26
0 30.7
0 35
0 29.3
0 28.2
1 25.8
1 27.5
1 25.2
1 27.2
1 28
1 27.3
1 29.8
1 28.7
1 28.7
1 33.4
1 32.8
1 33.8
1 30.8
1 32.2
1 33.7
1 25.5
1 25.4
1 28.8
1 26.3
1 26.8
1 32.1
1 27.9
1 30.1
1 31.6
1 29.5
1 33.6
1 31.8
1 29.1
1 32.4
1 39.5