Thanks to all.. for their useful disscussion

How about this that I drew with the Presentations Application that I sell through my web site. I essentially draw on a blank sheet of paper and use custom X and Y tick scales. The list plot results are scaled and shifted into position on the paper. I used Eric's data.

<< Presentations`
data1 = {{1, 0.0385}, {2, 0.04}, {3, 0.04096}, {4, 0.041044}, {5, 0.0408417}, {6, 0.0405}, {7, 0.04011}, {8, 0.03965}, {9, 0.0391528}, {10, 0.03861}};
data2 = {{1, 1.0386}, {2, 1.04029}, {3, 1.04108}, {4, 1.04113}, {5, 1.04091}, {6, 1.04056}, {7, 1.04015}, {8, 1.03969}, {9, 1.03918}, {10, 1.03864}};

The following draws the graphic. This has the advantage of a single x axis and it was easy to add Text labels.

Draw2D[
 {ListDraw[data1, Joined -> True] // 
    ScaleOp[{1, 1/0.004}, {0, 0.038}] // TranslateOp[{0, -0.038}],
  ListDraw[data2, Joined -> True] // 
    ScaleOp[{1, 1/0.004}, {0, 1.038}] // TranslateOp[{0, 0.2}],
  Text[Style["data1", 14], {6, 0.8}],
  Text[Style["data2", 14], {6, 2}],
  XTickLine[{0, 10, -0.1}, {0, 10}, {0, 10, 2}, 5],
  YTickLine[{0, 1, 0}, {0.038, 0.042}, {0.038, 0.042, 0.002}, 2, 
   YTextSpecs -> {{1, 0}, {1, 0}, 0.008},
   YNumberFunction -> (If[# == 0, 0, NumberForm[#, {4, 3}]] &)],
  YTickLine[{1.2, 2.2, 0}, {1.038, 1.042}, {1.038, 1.042, .002}, 2, 
   YTextSpecs -> {{1, 0}, {1, 0}, 0.008},
   YNumberFunction -> (NumberForm[#, {4, 3}] &)],
  Text[Style["Split Plot with Separate Scales", 14, Bold], {5, 2.3}]
  },
 AspectRatio -> 1,
 PlotRange -> {{-1, 10}, {0, 2.4}},
 PlotRangePadding -> 0.5,
 Frame -> None,
 ImageSize -> 350]

Thanks Simon.. But Method does not gives the accurate figure.. Because In this Way Both the Graphs of List1 and LIst2 looses their original shape.

Show[{
  ListPlot[List1, Joined->True], 
  ListPlot[Map[# - {0, .997} &, List2], Joined->True]
 }, PlotRange->All, Axes->{True, False}]

Look up each of those words in the help system and study them until you can understand how and why this was done.

You are free to draw independent axes if you wish

o = .997;(*Offset to upper plot*)
Show[{ListPlot[List1, Joined -> True],
  Graphics[{Text["0.039", {.5, .039}], Text["0.040", {.5, .040}], Text["0.041", {.5, .041}]}],
  Graphics[Line[{{.7, .039}, {.9, .039}, {.8, .039}, {.8, .040}, {.7, .040}, {.9, .040}, {.8, .040},
     {.8, .041}, {.7, .041}, {.9, .041}}]],
  ListPlot[Map[# - {0, o} &, List2], Joined -> True],
  Graphics[{Text["1.039", {.5, 1.039-o}], Text["1.040", {.5, 1.040-o}], Text["1.041", {.5, 1.041-o}]}],
  Graphics[Line[{{.7, 1.039-o}, {.9, 1.039-o}, {.8, 1.039-o}, {.8, 1.040-o}, {.7, 1.040-o},
     {.9, 1.040-o}, {.8, 1.040-o}, {.8, 1.041-o}, {.7, 1.041-o}, {.9, 1.041-o}}]]
  }, PlotRange -> {{.25, 10}, {.035, .045}}, Axes -> {True, False}]

Based on responses from other users who have been dissatisfied with the built-in graphics functions, I suspect you may not find this acceptable either. If that is the case then you might consider creating two separate plots, exporting those, importing those into a high-end publishing product and using it to produce the image that you desire. Someone wrote "if it takes x time to get the data and mathematics almost correct then it usually takes between 3x and 10x to get the graphics almost correct and 10x to infinity to get the graphics exactly the way you want them."