Is there any way to make a chart matrix like this (tried scatter, bubble, and radar)?

Short, useless, optimistic answer -- yes, it is possible to do this in Excel.

I am not certain I understand exactly what you are trying to do or exactly what you have tried. My first instinct is that your chart looks like a bubble chart, but you indicate that you have already tried a bubble chart, so I don't know what to recommend. I would be curious exactly what you tried when you tried using a bubble chart. I would have expected something like:

1) in the chart's data table, the first column would calculate the horizontal position of each data point.
2) the second column would calculate the vertical position of each data point.
3) the third column would calculate the size of each data point (I assume you know the formulas you need for each of these columns).
4) Select those three columns -> insert bubble chart.
5) Format the chart as desired.
6) Add data labels or text boxes at each corner to show race/gender for each corner of the chart.

What did you do differently when you tried a bubble chart? What more do you need this bubble chart to do?

A bubble chart needs three numeric columns -- It doesn't know what to do with text strings. If you selected those three columns, then it failed because it doesn't know what to do with the "male", "female", etc. text strings. You cannot use those three columns, so you need three new columns that effectively converts those three columns into something numeric.

This is probably not exactly right, but it should at least show the steps:

1) I need a column of values for horizontal position. Your chart shows male to the left and female to the right, and the axes suggest that the values extend from (-)10 to (+)10. Assuming your picture starts in A1 at the upper left, D2 might be =IF(B2="Male",-1,1)*A2 and copy down. This should return a negative hourly wage value for males and a positive wage value for females. I don't know exactly how you want gender to figure into horizontal position, so replace this calculation with whatever makes sense to you.
2) I need a column of values for vertical position. Your chart shows non-white at the bottom (at -10 again) and white at the top (+10). In E2, I enter a similar formula =IF(C2="White",1,-1)*A2 and copy down. Again, replace this formula with whatever makes sense to you for the vertical position.
3) Size, I assume is just a copy of the hourly wage in column A. To simplify creating the chart, I will make a copy of column A in column F =A2 in F2 and copy down.
4) Creating the chart should be as simple as selecting D2:F5 and inserting a bubble chart.

I don't understand. Your picture in the OP only shows one data series, so I'm not sure what you need as far as additional data series. With a bubble chart, you will add additional y/vertical position and size columns for each data series that you want to show (as well as an additional x/horizontal position column if the additional series will use different values for the horizontal position). Can you help us understand what you are wanting to do with the additional data series?

Attachments are invalid. Can you try attaching them again (make sure you are adding them from the manage attachments window)?

I cannot open any of the images. The picture(s) in post #1 are the only pictures I can see.