How much space does each user get for their mailbox with Office 365? What is the largest attachment size that they can send?

For the majority of Office 365 plans, each user gets 50 GB. The exceptions are the Kiosk plans (which get 2 GB) and Exchange Online Plan 2, E3, and E5 (which include an unlimited archive mailbox).

File attachments are limited to 25 MB in size and 250 files.

For full details on all the limits of Exchange Online, you can visit Microsoft’s site.