In your string, "\\" is an escape sequence that turns into "\":

my $DIR;
$DIR = '\\ax-risvis447\c$\calc';
print $DIR;  # prints "\ax-risvis447\c$\calc"

# This gets you the string you want:
$DIR = '\\\\ax-risvis447\c$\calc';
print $DIR;  # prints "\\ax-risvis447\c$\calc"

# And if you want to use double quotes, escape the "$" too:
$DIR = "\\\\ax-risvis447\\c\$\\calc";
print $DIR;  # prints "\\ax-risvis447\c$\calc"
[download]

However, you may still have issues due to shell escapes sometimes being tricky on Windows. For a "cleaner" way to list the files in a directory, see this recent thread for several approaches.

If you still have problems - what happens when you type in dir \\ax-risvis447\c$\calc on the command line?

As a general side note, you should be using use warnings; use strict; at the top of your scripts, to help avoid some common mistakes and typos.