To create hatched rectangle primitives we can use the function hatchF (from this answer and this answer to closely related questions).

ClearAll[hatchF] hatchF[mf_List: {# &, #2 &}, mesh_List: {50, 50}, style_: GrayLevel[.5], opts : OptionsPattern[]] := ParametricPlot[{x, y}, {x, 0, 1}, {y, 0, 1}, Mesh -> mesh, MeshFunctions -> mf, MeshStyle -> style, BoundaryStyle -> None, opts, Frame -> False, PlotRangePadding -> 0, ImagePadding -> 0, Axes -> False] 

Use hatchF to create two Graphics objects

t1 = hatchF[{# - #2 &}, {30}, White, MeshShading -> {GrayLevel[.8], GrayLevel[.5]}]; t2 = hatchF[{# - #2 &, #2 + # &}, {30}, Directive[Thick, White], MeshShading -> {{GrayLevel[.6], GrayLevel[.8]}}]; 

and take an image from ExampleData[]

t3 = ExampleData[{"ColorTexture", "Metal4"}]; 

We can use Texture[t1], Texture[t2], etc. as directives for styling individual data elements using Style[element1, Texture[t1]], Style[element1, Texture[t1]] etc. Together with the built-in (but undocumented) function System`BarFunctionDump`TextureBar used as the setting for ChartElementFunction we get the desired styles for selected data elements.

BarChart[{{Style[0.25, Texture@ t1], 0.4, 0.15}, {0.172, Style[0.25, Texture@t3], Style[0.18, Texture@t2]}, {0.101, .26, 0.17}}, BarSpacing -> {0.15, 1}, ChartLegends -> {Text[Style["BXE", 12]], Text[Style["CXE", 12]], Text[Style["TQ", 12]]}, ChartStyle -> "GrayTones", LabelingFunction -> Above, ChartLabels -> {{"Large", "Small", "Overall"}, None}, BaseStyle -> Directive[FontSize -> 11], PlotLabel -> Style["Volume", FontSize -> 15], ChartElementFunction -> System`BarFunctionDump`TextureBar] 

gives the picture above.

To create hatched rectangle primitives we can use the function hatchF (from this answer and this answer to closely related questions).

ClearAll[hatchF] hatchF[mf_List: {# &, #2 &}, mesh_List: {50, 50}, style_: GrayLevel[.5], opts : OptionsPattern[]] := ParametricPlot[{x, y}, {x, 0, 1}, {y, 0, 1}, Mesh -> mesh, MeshFunctions -> mf, MeshStyle -> style, BoundaryStyle -> None, opts, Frame -> False, PlotRangePadding -> 0, ImagePadding -> 0, Axes -> False] 

Use hatchF to create two Graphics objects

t1 = hatchF[{# - #2 &}, {30}, White, MeshShading -> {GrayLevel[.8], GrayLevel[.5]}]; t2 = hatchF[{# - #2 &, #2 + # &}, {30}, Directive[Thick, White], MeshShading -> {{GrayLevel[.6], GrayLevel[.8]}}]; 

and take an image from ExampleData[]

t3 = ExampleData[{"ColorTexture", "Metal4"}]; 

We can use Texture[t1], Texture[t2], etc. as directives for styling individual data elements using Style[element1, Texture[t1]], Style[element2, Texture[t2]] etc. Together with the built-in (but undocumented) function System`BarFunctionDump`TextureBar used as the setting for ChartElementFunction we get the desired styles for selected data elements.

BarChart[{{Style[0.25, Texture@ t1], 0.4, 0.15}, {0.172, Style[0.25, Texture@t3], Style[0.18, Texture@t2]}, {0.101, .26, 0.17}}, BarSpacing -> {0.15, 1}, ChartLegends -> {Text[Style["BXE", 12]], Text[Style["CXE", 12]], Text[Style["TQ", 12]]}, ChartStyle -> "GrayTones", LabelingFunction -> Above, ChartLabels -> {{"Large", "Small", "Overall"}, None}, BaseStyle -> Directive[FontSize -> 11], PlotLabel -> Style["Volume", FontSize -> 15], ChartElementFunction -> System`BarFunctionDump`TextureBar] 

gives the picture above.