To be able to backup databases to a network share, the MSSQLSERVER service needs access to the location on the network.
For this you need to change the SQL services service account.
Before doing this you'll need to give that user extra permissions, to accomplish this you can add the user to all the SQLServer2005* or SQLServer2008* groups.
To change the logon user go to "SQL Server Configuration Manager". Under Services, click on properties for the MSSQLSERVER.
Here you can change the logon account to the one you'll need. Specify this to a domain user, or a local user that also exists on the machine where you will store the databases, and that has the full access share and NTFS permissions.
After modifying the logon name, you'll have to restart the service. Do the same for the SQL Server Agent.
Now you'll be able to backup databases to a network share and specify the path like this then: \\servername\share.