The Cookbook suggests Sys::Hostname then gethostbyname (or the POSIX module's uname()):

use Sys::Hostname;
my $hostname = hostname();

my $ip = gethostbyname($hostname);
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(Untested, but pretty much verbatim from the book.)

The POSIX::uname() method looks like:

use POSIX qw( uname );

my $hostname = (uname())[1];

# the rest as above
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On systems that don't implement any part of POSIX, you'll have ... other... results. You might also try reverse resolving any IP address you get with gethostbyaddr.