Q: Can I use my mac as a time machine backup for another mac on the same ethernet network

Yes, you can do that, but you must first purchase and install OS X Server.

I would like to discourage you from doing that.  How do you propose to backup the older mac?  An external dedicated HDD  is the best option and one disk could backup both macs.

Ciao.

You'd be better off removing the old Mac's hard disk drive and installing it in a $10 enclosure...

Yes, you can use a folder shared from a Mac as a Time Machine backup destination. You don't need OS X Server. The folder must be shared via AFP, not SMB. The clients must connect to it with the login credentials of the user who owns the share. You didn't ask whether it's a good idea, so I'm not addressing that issue.

It's not a good idea to mix backup and non-backup data on the same drive. That might not be a problem if you intend to use the old Mac only as a backup destination. Regardless, you always need more than one backup to be safe, and network backups are inherently slower and less reliable than backups to a locally-attached storage device.