There was a glitch that allowed you to friend your alts but IDK if they fixed it or not, it involved DCing while going into an instance or something like that, you can google it if you want. Pulling it off lets you send mail to your alts.

IMO there's really no excuse to not let you mail things between your alts on the same server or add them as housing tenants. It really feels like SE made the game assuming nobody would ever want more than one character and that even letting you have alts was an afterthought.

Aside from the above glitch if it still works, your options are pretty much:

A. Use the MB to xfer gil by selling cheap stuff for high amounts. Or to xfer items but you'll need to be careful with how you price things so someone doesn't buy it while you're swapping characters.
B. Use a FC chest to xfer gil and items. This assumes your characters are in the same FC and have access to the fc chest to freely remove items.
C. Get a friend you trust to hold onto stuff while you relog.