Would you assume that every equation you write down hs an analytical solution? Why would you?
First, let me write the equation as an expression, instead of an equality. This way we can plot it, and then look for a zero crossing.
eq = -L + ((g)*(T)^2/(2*pi)) * tanh((2*pi*h)/L);
And what we see is a "function that crosses y==0 around L == 0, but as you approach L == 0, you divide by L. Therefore you have a singularity at L==0, exactly where the curve crosses zero.
No solution exists, because your expression is undefined at L == 0. So no, you cannot fix this. Nor can you solve for it. This is not a question of you not understanding how to use solve properly. It is a question of applying solve to something where no solution exists.
When something strange happens, PLOT IT!!!!!!!! Plot everything! Then when you have plotted everything you can think of, try plotting something else. And think about what you see there.