How to share a very large voice memo to Dropbox from iPod?
Since you only have two sharing options in Voice Memos (Email, Message), you are obviously limited to those two things (clearly excluding a computer which you don't have access to). 

Caveat: the following steps worked for me, but I didn't have a super-large voice memo to test it with.


Turn off "Send as SMS" in Settings.app -> Messages (if enabled).
Open Voice Memos.app and select the memo you want to share.
Tap "Share", and select "Message".
Type in a bogus number.

This shouldn't be hard because we just limited our playing field to only iMessage users by turning off SMS alternate. The goal is to not reach someone obviously.

Hit send.
You should receive an error message that the message failed to send. (Perfect.)
Tap the memo, and then select the "Share" icon in the upper right corner.
Select "Open in DropBox".