I wanto sort @list. But without using any Date Sort modules. But we I can use perl's sort function.

I find this mostly annoying. Every now and again someone pops up and asks "how can I do XYZ without using any module?" In most cases this is plainly stupid - check e.g. this quotation, and is being asked because the person who asks does not have root privileges and ignores that modules can be installed locally too. See for example this section of the Tutorials.

In this case your post may suggest you're asking about homework instead, which is also advised against here. Personally I don't find anything wrong with it, provided that you clearly state so and you show what you've tried, so that we do not give ready-made solutions, which most of us won't do in any case -because we're not a helpdesk- and help you to learn instead.

Please don't use split also.

This also smells too much like homework, which I don't like. Indeed, if you have your numbers e.g. in a "MM-DD-YYYY" format as some of your post seem to imply, then it is quite natural to get individual chunks to sort upon like thus:

my ($m, $d, $y) = split /-/;
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you don't like it? Then

my ($m, $d, $y) = /\d+/g;
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will work just the same if you trust the format of your dates. Now, is this acceptable for your teacher?