runrig and AndyZaft have given you better ways of doing what you want. I'd just like to tell you why your method doesn't work.

Backticks capture the output of an external command. In list context (as you have) they return a list of lines, and each of those lines ends in a linefeed (on *NIX platforms). You need to remove them using chomp:

my @files = `ls Data_files/*/Data.txt`;
chomp @files;
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or

foreach my $filename(@files){
    chomp $filename;
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