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I have an Excel spreadsheet that has columns A-U and rows 1-50. 
When I print it at a readable scale, it prints in a three-page-wide format:
page 1 is columns A-F wide and rows 1-20 long,
page 2 is columns G-L and rows 1-20, and page 3 is columns M-U × rows 1-20. 
Then pages 4, 5, and 6 go to rows 21-40, and so on. 
I don't want to scale the printout to be 1 page wide and n pages tall
because that would make the cells too small. 
I would rather have a one-sheet-wide format
in which columns A-U of one row of the spreadsheet
print using two or three rows on the printed page
but keeping the Excel table format, if possible,
and then the next row of the spreadsheet, and so on.

How can I do this?

Use the page setup settings to scale the printout to be 1 page wide and n pages tall. The exact look of the dialog may depend on your printer.

One way to set this is: Go File > Print and select the Scaling dropdown under the Settings. Select Custom and on the Page tab set "Fit to" 1 page wide and 99 (or any large number) tall.

Edit: what you describe in your latest comment is not possible. Excel prints the grid. It can't re-arrange single rows and break them across to fit on one page.

Without using an "export to external program" option, you have two choices, each with their drawbacks. Both use the File ⇨ Save As.. dialog

Option one, save as "Formatted Text (Space delimited)". This will create a .prn file which you can rename as a .txt file. It creates nice columns of data that is reasonably clean. The problem is that the line length is limited to 240 characters (IIRC), and then the lines are continued. This is odd behavior in that the continuation of the first line will be started after the first part of the last line, not wrapped directly below the first line. So with 80 lines in the sheet, line 81 would be part 2 of line 1, and 161 would be part 3 of line 1.

Option two, save as "Text (MS-DOS)". This will create a .txt file, which you may not see the extension of if you have "Hide known extensions" turned on. Even if the extension is hidden you'll know it's a normal text file by the icon, and Windows Explorer will call it a "text file" in the "Type" column. The problem with this option is that if the columns vary much in size it gets rather sloppy. That's the size of the data in the column, not the size of one column compared to another. If all the data in a column is nearly the same length, then it will still line up well. This option does not have an imposed line length limit. Notepad, however is limited to 1024 characters per line, IIRC. This isn't likely to be a problem for your case, since it will wrap the lines in a normal fashion, and you intend to do that anyway.

After saving according to either of the two options above, and changing the extension if needed for option one, open the file in Notepad. Turn on word wrap, if it is off, by selecting Format ⇨ Word Wrap on the menu. Now you can print it from Notepad, controlling the header, footer, and margins as you see fit.

If your dataset has a column with widely varied content lengths, and if your usage of the printouts allows, you should try to move that column, or columns, to the far right of the spreadsheet. Columns like description, address, last name, product name, etc. are candidates for such a move. Dates, prices, phone numbers, and often product codes, are usually fairly consistent in length, and therefore, work best near the front of the line.

It is possible to do something like what you want. 
It’s a lot easier if all your columns are the same width,
and you want to put the same number of columns (e.g., 7; i.e., 21÷3)
on each row. 
If that’s acceptable to you,
and assuming that you have your data on Sheet1, A1:U50, then enter

=OFFSET(Sheet1!$A$1, INT((ROW()-1)/3), COLUMN()-1+7*MOD(ROW()-1,3))

into Sheet2!A1, drag/fill to the right to Column G
(i.e., the 7th column), and drag/fill down to Row 150 (3×50). 
Cells Sheet2!A1:G1 will display Sheet1!A1:G1. 
(You may need to do some work to get it formatted correctly.) 
But then Sheet2!A2:G2 (Row 2) will display Sheet1!H1:N1,
and Sheet2!A3:G3 (Row 3) will display Sheet1!O1:U1. 
And Sheet2 Rows 4-6 will display Sheet1 Row 2, and so on.

For example, this Sheet1:

yields this Sheet2:

Then you just print Sheet2.

^{This is patterned after my answer
to How to split one column in Excel into multiple columns of 55 rows.}