I agree with Peter that there is no contradiction the t test and the correlations are telling you very different things. First of all as Peter mentioned statistical significance depends on both the magnitude of the differnce and the sample size. With a very large sample size small difference can be significant (even highly statistically significant).
Now correlation between the variables measures whether or not the move together in a linear fashion. It may be that if you have paired data for X$_1$ and X$_2$ that they don't tend to increase of decrease together or in the sense of negative correlation have X$_1$ decreasing while X$_2$ increases. So for your data the magnitude of this tendency is low or non existent.
Now two variable can have very different means and zero correlation such as if
X$_1$(k)= A + ε(k) and X$_2$(k)= B + η(k) where the ε(k) and η(k) are uncorrelated 0 mean noise terms that are also uncorrelated with each other.
Suppose A-B>0. Then for sufficiently large n (how large n has to be depends on the magnitude of the difference between A and B) the t test will say that the mean of X$_1$is statistically significantly different from the mean o9f X$_2$. But X$_1$ and X$_2$ are uncorrelated. This is like your situation.
On the other hand suppose X$_1$(k) = X$_2$(k) + ε(k) where the ε$_s$ are zero mean independent gaussian noise terms their means will be the same but they could be highly correlated with the degree of correlation dependent on the variance of ε(k).