'no strict "vars"' would obviously shut strict up, but I would still like strict's variable name checking.

Since strict vars relies on lexical analysis, you have a perverse need to use eval() and you already mentioned that you don't know which variables will be used (how is this in any way realistic again? This seems like a prime example of an XY Problem) you have no choice but to disable strict vars. Luckily you only need it at the eval point:

sub eval_this {
      my $code = shift;
      $code = "package main; \n$code";
      no strict 'refs';
      eval $code;
      die ("Error: $@") if $@;
}
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update: you could do:

eval_this('our $var; print $var');
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Which doesn't really improve on the situation at all, but would shut up strict.