Even a TiVo will work that way....
You have seen the ads "Only __(Our company)___ lets you record in one room, play back in another, or take it with you anywhere you want to go". (There are a few more features too).
Well, The program I'm watching on DVR Front, I recorded on DVR Rear, Excess recordings for times when I have poor TV coverage,,,, Stored on the hard drive of this and one other computer.. I can control the DVR's from this or other comptuers, or my android phone, or tablet or.. Well it goes on and on.
Any good "Can take it with you" dvr, coupled with an external HD for your computer (Or a very large internal one) and a computer with HD out will do it
Record at home, copy to computer to Take it with you, and off you go.
Only negative about mine, they are "Vintage" no longer made Analog (NTSC) recorders so I use a digi-verter to get ATSC broadcast.. Or they can record analog cable direct.
WHat's more.. They can do both under computer control (Record 8-9 on Cable, 9-10 on Antenna (Digital over the air converter) and go back to Cable for 10-11 if that's the way the shows are shown...
And they do oh so much more as well.. REPLAY TV with WiRNS running on a dedicated (Small) computer.