Can I make an ubuntu network and have users be able to access accounts on multiple machines.
Yes, you can. Please look up how to set up openldap to use LDAP to distribute password file, group file etc to all your machines. You could also use Kerberos to get singel sign on to all the machines. Actually, Microsoft has based AD on those tools, and utses CIF (smb) to distribute config files and users files.
You would also want to distribute your users home directories too. You could use NFS for that. You might want to use NFS in combination with autofs so the users home directory get mounted when needed.
So yes, look at openldap and/or kerberos and NFS/autofs to get what you asked for.