How to take SQL Server database under this scenario? I have a database of size 290 GB running on SQL Server 2005 (compression is not available) and disk space is z: drive 250 GB and Y: drive is 80 GB. 

How can I take a full backup in this scenario? How can I split full backup into different files in this ratio? 

Please suggest a method to do this.
If you have access to another server on your network with larger disks, you can do database backup to a network share:

backup database YOURDATABASE 
to disk = '\\NETWORKSERVER\SHARENAME\YOURDATABASE_yyyymmdd.bak' 
with init


For this to work, the share would need to have Write permissions from the domain account under which your SQL Server is running.

If you don't have access to another server with enough disk space, you may want to get an inexpensive Network Attached Storage type of device (e.g. synology) and perform your backups to it.