Not usually. Plus, the name "mainframe" isn't used much anymore. There are people that will have servers in their homes, either to run as a firewall, local mail server, or to maintain files, but that is fairly rare.
If you have a lot of data, most people will purchase CDs or DVDs and burn the data to them for safe keeping. Others will get USB flash drives, or even USB external hard drives. That way the data can be both stored and removed if not needed anymore.
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