Is it possible to compute a correlation between two variables from just means, standard deviations and sample sizes? Is it possible to compute the correlation between two variables with just the mean, standard deviation, and sample size?
If so, how would I go about it?
 A: It's not possible to compute correlation with just information about the individual variables.
The correlation is a particular measure of how they vary "together". Information like mean and standard deviation is how they each behave on their own, without any consideration of other variables.
(Specifically, correlation is a property of their joint distribution, while the quantities you mention are properties of the marginal distribution, and will be consistent with different joint distributions.)
In the diagram below, the populations y and z were drawn from have the same mean and standard deviation:

Yet as you see, their correlations with x are quite different.
If I only told you the means and standard deviations and n's, you would have no way to distinguish whether I meant the first plot or the second plot (or indeed, any number of other possible plots).
A simple analogy is to think of a two dimensional table, with totals at the bottom and the right edge. If you only look at the row and columns totals, you can't figure out much about which cells in the interior of the table contributed the values. So these margins:
                    Total
        ?   ?    ?   10
        ?   ?    ?   20
        ?   ?    ?   15
 Total 10  20   15

are consistent with all of these tables:
 10   0    0         2   5   3          0   5   5
  0  20    0         5   8   7          5   5  10
  0   0   15         3   7   5          5  10   0

