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".