Do you know if it is possible to download SAP R/3 a s a free/trial version for running it on a private laptop?
SAP is not free software so you cannot download and use it. You can use an IDES version for training and learning but again you need a valid OSS ID and a license that is only available for their Customer and Partner.
The trial version available on the SDN download section is only useful for technical training e.g. Abap, Basis. If you want to learn a functional module, then you need an IDES.
Regards,
Subhash