A two-dimensional Ising square lattice is modeled as a nano-size block array to study by Monte Carlo simulation the magnetic thermal stability of nano-structure magnetic media for data storage, thereon in the blocks J
1 > 0 is assigned for the interaction of a pair of nearest-neighbor spins, while 0
J
0 J
1 for that in regions between the blocks and (J
0 + J
1)/2 for the nearest-neighbor pairs with one in the block and the other one out of but near-most the block. We show that the magnetic thermal stability of the block accrues with the increase of J
1 and with the decrease of J
1 - J
0 for a given J
1, but contrarily, the anchoring ability for the initial magnetic orientation in nano-size block trails off as J
1 - J
0 diminish. This phenomena and size dependence of such anchoring ability are discussed in detail.