You tried to find analytical solution the turning it to numbers N[DSolve... better above or:
In[10]:= sol =
NDSolve[{A'[t] == 0.5/((1 + (B[t]/.5))^10) - 0.2 A[t],
B'[t] == 0.5/(1 + (A[t]/2))^10 - 2 B[t], A[0] == 0,
B[0] == 0}, {A[t], B[t]}, {t, 0, 120}]
Out[10]= {{A[t] -> InterpolatingFunction[{{0., 120.}}, <>][t],
B[t] -> InterpolatingFunction[{{0., 120.}}, <>][t]}}
In[11]:= Plot[{A[t], B[t]} /. sol, {t, 0, 120}]
