As far as I know, you can't [Added: There is a way, see my other reply below, You *Can* Catch errors in closing lexical filehandles]. Observe,

# use Fatal qw/open print close/;
use Fatal qw/open close/;
my $result = eval {
    open my $fh, '>', '/dev/full';
    print $fh "Foo\n" or die $!;
};
print $result ? $result : $@, $/;
[download]

prints '1', the result of printing to the filehandle.

Placing an explicit close after the print statement gives something like,

Can't close(GLOB(0x804b548)): No space left on device at (eval 2) line
+ 3
    main::__ANON__('GLOB(0x804b548)') called at -e line 1
    eval {...} called at -e line 1
[download]

That uses the handy /dev/full device of linux, for which writes always fail with ENOSPC. The failure would have occurred on print if we had set $fh to autoflush or had printed more than a buffersworth of text.

For careful error handling, you should close your lexical handles as if they didn't know how to do it for themselves.

Update: gaal++ points out an error. Corrected.